Der SV Werder Bremen kehrte vom Auswärtsspiel gegen FC Bayern München mit leeren Händen zurück. Am Ende hieß es Im Hinspiel hatte sich Bayern. 1. Dez. Ein Tor machte den Unterschied in der Begegnung vom SV Werder Bremen ( Gesamtmarktwert ,9 Mio. €) und FC Bayern München. 1. Dez. Bundesliga Spielbericht für Werder Bremen vs. Bayern München.
You will find what results teams Werder Bremen and Bayern Munich usually end matches with divided into first and second half.
Last 6 matches stats 6. Last matches Werder Bremen. Nuernberg - Werder Bremen. Werder Bremen 2 - 2 Eintracht Frankfurt. Hannover 96 0 - 1 Werder Bremen.
Werder Bremen 1 - 1 Hoffenheim. Borussia Dortmund 2 - 1 Werder Bremen. Last matches Bayern Munich. Bayer Leverkusen - Bayern Munich.
It was indeed a beautiful goal after a fantastic pass from James and a solo that was reminiscent of the old Frank. Frank Ribery is fighting for a contract extension at Bayern.
There were plenty of doubts when Werder Bremen signed Jiri Pavlenka at the beginning of the season to replace homegrown player Felix Wiedwald.
Bayern vs Werder Bremen — Sunday, January 21, The result means that Bayern are now points ahead of second-placed RB Leipzig and with the hunting pack happily taking points from one another Bayern are likely to increase the lead at the top of the table further.
Furthermore, in Werder Bremen, they are not only facing a side currently battling against relegation but a team that has lost the last 14 encounters against the Rekordmeister.
With all of this in mind, almost nothing can go wrong for Bayern, who are now just counting down the matchdays until they can put a blue tick next to the 28 th Bundesliga title.
Another treble would not only be historical, no club around Europe has managed to win a treble twice, but would also be sensational given the mood at the club in the later stages of the Carlo Ancelotti era.
Werder Bremen in the meantime are more focused on preserving their Bundesliga status. Throughout the late s, s and early s Werder were the counterweights to Bayern in German football.
Borussia Dortmund 2 - 1 Werder Bremen. Last matches Bayern Munich. Bayer Leverkusen - Bayern Munich. Bayern Munich 4 - 1 VfB Stuttgart.
Hoffenheim 1 - 3 Bayern Munich. Eintracht Frankfurt 0 - 3 Bayern Munich. Latest matches with results Werder Bremen vs Bayern Munich.
Werder Bremen 0 - 1 Bayern Munich. Bayern Munich 5 - 0 Werder Bremen. Bayern Munich 2 - 0 Werder Bremen.
Bayern Munich 6 - 0 Werder Bremen. Werder Bremen 1 - 2 Bayern Munich. Werder Bremen 0 - 2 Bayern Munich. Bayern Munich 4 - 2 Werder Bremen.
Statistics of the season All Home Away. Go to the league. Max Kruse vs Hanno Behrens. Show schedule for teams Werder Bremen and Bayern Munich.


Create exciting school projects or cool promotional materials in seconds. This Oslo logo can be used in endless contexts. Just a sample of use: greetingcards, banner Ads, folders, giveaways, stationary, magnets and ads. Creating logos can not be any easier?
Our Oslo logos can be used for whatever you need. You can use them at work, home or school projects. Just a small example of usage: sales materials, name tags, YouTube videos, email advertising, gift tags, websites and photo posters. All logos can be easily customized to your needs!
These Oslo logos you can use for all occasions. Both for personal or professional use. For example you can use them to create: giveaways, image processing, theme parties, internet Marketing, calendars, posters and business cards. All logos can be easily customized to your needs!
Our Oslo logos can be used for whatever you need. You can use them at work, home or school projects. Just a small example of usage: christmas cards, spreadsheets, documents, email advertising, stickers, birthday card and promotional materials. All logos can be easily customized to your needs!


Robert Vadra is back in the headlines after a report by Haryana IAS officer, Ashok Khemka, found multiple violations in land deals by Sonia Gandhi's son in law. Those in defence of Mr Vadra claim there was nothing illegal about these deals. But a closer look at these transactions suggest that they walk a fine line between illegality and outright cronyism, where property traders, Congress politicians, state governments and business houses do extraordinary favours to Vadra to ensure that he becomes a multi-millionaire without spending a penny.


text:
Linkin Park Write a Heartbreaking Open Letter to Chester Bennington Following His Suicide
By Jonathan Borge Updated Jul 24, 2017 @ 12:15 pm
Less than a week after 41-year-old Linkin Park singer Chester Bennington committed suicide by hanging, the group's members have come together to release an official statement.
On Monday, Linkin Park released an open letter directly addressed to Chester on their official website and Facebook.
"Dear Chester," it starts. "Our hearts are broken. The shockwaves of grief and denial are still sweeping through our family as we come to grips with what has happened."
In the letter, band members address the support they've received from fans and also speak for Talinda, Chester's wife. "Talinda and the family appreciate it, and want the world to know what you were the best husband, son, and father; the family will never be whole without you," the band—whose remaining members include Mike Shinoda, Rob Bourdon, Brad Delson, Dave Farrell, and Joe Hahn—collectively writes.
Additionally, the letter speaks to Bennington's "infectious" spirit and the "demons" that led to his suicide. "We're trying to remind ourselves that the demons who took you away from us were always part of the deal. After all, it was the way you sang about those demons that made everyone fall in love with you in the first place."
In the letter, the group also leaves their future open-ended: "Our love for making and performing music is inextinguishable. While we don't know what path our future may take, we know that each of our lives was made better by you. Thank you for that gift. We love you, and miss you so much."
Since Bennington's death, Live Nation announced the cancelation of Linkin Park's North American One More Light tour.
RELATED: Celebrities Respond to Linkin Park Singer Chester Bennington's Tragic Death
Our thoughts are with the brand and Bennington's family and friends.

redpajama_set_name: RedPajamaCommonCrawl


I am new here, What should I expect when I visit?
Welcome to the website of Faithway Baptist Church, located across from Ypsilanti High School on the west side of Ypsilanti, Michigan. We are an independent, Baptist church with a conservative style of worship and a strong emphasis on sound Bible preaching and teaching.
We exist to spread God's fame for the sake of His name by making and maturing disciples through the ministry of His Word. We would like to extend a personal invitation for you to join us as we seek to spread God's fame.
This mission statement contains our three core objectives: The exaltation of God, the evangelization of the lost, and the edification of the saved. These three core objectives are the basis of our guiding principles.
Thank you for taking a look at Faithway Baptist Church. Please contact us directly by email: faithwaybaptist@comcast.net or call (734) 482.8282 for any questions you might have. We extend a warm invitation for you to join us here on the west side of Ypsilanti at each and every opportunity.
Our Mission: Spreading God's Fame..
We have all sinned. Sin separates us from God, but God sent His Son, Jesus Christ, to die on the cross to pay for the sins that we have committed. If you will receive His gift, you can know that your sins are forgiven, and that you have a place prepared for you in heaven.
We believe the Bible is the Word of God and is without error.
We believe Jesus Christ is the Son of God, virgin born. He died on the cross and offers to us a free gift of eternal life through His substitutionary atonement (that is, taking the punishment we deserve).
We believe He will return for His own at the rapture and then will come to this earth again to set up a thousand year kingdom.
We believe in a literal heaven and hell and know that those with Christ will be saved, and those without Christ will spend an eternity in the lake of fire.
We also believe that those who receive Christ are eternally secure and cannot lose their eternal life.


Heidi lives with Dea in Texas and she is taking the show scene by storm. After three weekends of shows, Hedi has 9 points including her majors towards her championship.
She is catching the eye of judges inside and outside of the ring!
harsh outer coat and a dense, waterproof pile for undercoat.
We are having a lot of fun showing and living with Heidi here in the Lone Star State!
Artwork and images may not be used or reproduced without written permission.


Famous blogger, Hilath Rasheed has claimed that he knows those behind the "Maldivian Atheists' Facebook page, which mocks Allah and the Prophets, and speaks against Sharia Law on Sunday.
Former journalist, Hilath made the statement in a post published on his online blog.
"I know who runs Maldivian Atheists Facebook page. I met them in Male'. I don't know their real names or addresses but I know their FB nicknames but I am going to keep it a secret even if the ISIS killed me", Hilath wrote.
He added that Sulthona, and missing journalist Ahmed Rilwan - the two who are being accused of being behind the disrespectful page are innocent.
Addressing the disappearance of the journalist and the accusations, the blogger said that it is a cheap trick used by the government to justify their stand in the case.
"Now "president" Yameen is using the same cheap trick: spread the lie that Rilwan was anti-Islamic and an admin of the Maldivian Atheists page and hey presto, most devote Maldivians find it acceptable that Rilwan should be kidnapped and killed", He wrote.
Writing further about the issue, Hilath accused Ismail Abdul Raheem, a spy for the Wahabbi-Salafis in the Maldives Police Force, of being the culprit of the kidnapping.
Hilath also posted a screenshot of a message that Rilwan had sent to him through Facebook back in the month of February. Rilwan had asked Hilath about information regarding the person who was found guilty of attacking him in Artificial Beach, and said that the same person had been stalking him for the past few days.


text:
Matt Bodo, Housing Advocate, to be Commencement Speaker
Foothill College students can achieve anything that they want to when supported by others. After a few rough quarters, Matt Bodo wasn't sure if he was fit for college. He now plans to transfer to UCLA to major in linguistics and psychology.
"I didn't expect to attend college when I finished high school. I didn't think that I would get my GED," Bodo said. "One of the faculty at the Adult Education Center convinced me to go to Foothill, and I figured I would try."
Bodo didn't immediately succeed in college. His first quarter's GPA was a 0.0. During his second year, he became homeless. This led him to discover the college's support services. With the right resources, Bodo flourished and became an incredibly active student.
"I really didn't think I could do it," Bodo said. "But I didn't want to give up."
Bodo has been a part of Extended Opportunity Programs and Services (EOPS), the Associated Students of Foothill College (ASFC), and has spent countless hours working to support homeless community college students.
He still finds himself shocked at where he's ended up. "I couldn't have ever pictured myself in the position I'm in today, and I know that I've become the person I want to be," he said.
Bodo hopes to inspire students experiencing homelessness to succeed. He thanks EOPS, ASFC and Foothill's faculty and staff for the guidance and support that helped him so much.
STORY TAGS:
#fhgrad19
communitycollege


text:
Home/Blackhawks Prospects
Meet The Prospects: Paul Ludwinski
By Jesse Courville-Lynch
2022 CHL/NHL Top Prospects Game / Chris Tanouye/GettyImages
Welcome back to Meet The Prospects. This article highlights the Blackhawks 2nd Round Selection in this past draft. At 39th overall, the team selected Paul Ludwinski from the Kingston Frontenacs of the OHL after he played 67 games, and put up 43 points. At 5'11, 172 pounds, Ludwinski has a speedy side to him, with a hockey IQ that was praised by many scouts going into the draft.
Ludwinski is 18, and was born in Toronto, Ontario. In the 2019-20 season, he played in Toronto and served as the captain of the Toronto Marlboros U16 AAA team, where he scored 15 goals and added 17 assists through 31 games averaging just over a point per game.
He didn't play a single game in the 2020-21 season but was owned by the Frontenacs, and the 2021-22 season was a good showing for the young gun. 43 points in 67 games with 20 penalty minutes is a really impressive rookie season, and was definitely one that caught the eye of the Hawks.
Ludwinski won't be ready for the NHL for a couple seasons but I would fully expect him to get an increased role in Kingston, while pushing for a point per game. I'd say by 2024, he's ready to try and crack the Blackhawks roster as a middle six forward. His speed and shot are only bonuses to his potential upside as a really smart playmaker.
He isn't at the top of the Blackhawks prospect pool by any means, but he's definitely someone any team would want to have developing in their system. He fills all the boxes as far as skill, heart, defensive game, goal scoring and more. I really like Ludwinski, and I'll be keeping tabs on him the entire season.


text:
Ukraine Liveblog Day 116: Major Government Operation in Mariupol
A Ukrainian soldier stands in front of a burnt out BRDM armoured reconnaissance car with the initials of the 'Donetsk People's Republic' on it. Photo: @euromaidan
Yesterday's liveblog can be found here. For an overview and analysis of this developing story see our latest podcast.
Please help The Interpreter to continue providing this valuable information service by making a donation towards our costs.
View Ukraine: April, 2014 in a larger map
For links to individual updates click on the timestamps.
Below we will be making regular updates. Be sure to check back often and hit refresh.
US State Department: Russian Tanks and Grad Rockets Crossed Border
The US State Department has confirmed that Russia has indeed sent both T-64 tanks and Grad rocket launchers across the border. The New York Times reports:
A convoy of three T-64 tanks, several BM-21 "Grad" multiple rocket launchers and other military vehicles crossed the border near the Ukrainian town of Snizhne, State Department officials said. Reports and images of the weapons' presence circulated on Thursday, but there were conflicting claims about where they had come from.
"This is unacceptable," said Marie Harf, the deputy State Department spokeswoman. "A failure by Russia to de-escalate this situation will lead to additional costs."
A Western official said that intelligence about the movement of the tanks and other weapons into Ukraine was shared on Friday with NATO allies.
As we reported earlier there was evidence that the Russian-backed separatists fired Grad rockets earlier, but the weapon malfunctioned (possibly because the crew was inadequately trained) and at least one civilian was killed in the aftermath. We also reported earlier that NATO calls the allegations of Russian armor crossing the border a "serious escalation," and we analyzed the evidence that tanks may have crossed over from the Russian side of the border into Ukraine.
Headed Toward A Russian Gas Cutoff
Ukraine's energy minister has called for a new round of talks as Ukraine's Prime Minister has told his government to start preparing for Russia to cut off Ukraine's natural gas. RFE/RL reports:
Ukraine has said it is ready to start settling its gas bills if Russia agrees to charge $326 per 1,000 cubic meters of gas for 18 months.
Russia has said its "final price" is $385 per 1,000 cubic meters and threatened to cut off gas supplies to Ukraine if Kyiv does not pay state-controlled Gazprom $1.95 billion by June 16.
Ukraine And Russia Both Claim The Other Has Violated the Border
The head of the Russian FSB press service has released a statement saying a Ukrainian BMP armored vehicle crossed the border today, making it to the Russian town of Millerovo, east of Lugansk (map).
Interestingly, Ukraine claims a Russian Mi-35 helicopter violated Ukrainian airspace today near Kuibysheve (map), which, if true, likely means the helicopter was in the vicinity of Mariupol, a town which was retaken by the Ukrainian ATO earlier today. And it should be mentioned that Ukraine claimed that three tanks crossed over the Russian border yesterday (see updates below).
Eight OSCE Monitors Still Held 'Hostage' By Russian-Backed Separatists
The US ambassador to the OSCE Daniel Baer has made statements that have condemned the holding of OSCE monitors as hostage by separatists in eastern Ukraine:
Mr. Chairman, we remain deeply concerned that separatist forces continue to hold hostage eight OSCE Special Monitoring Mission monitors that they abducted on May 26 and May 29. That's two weeks ago today, and 17 days ago today. We condemn in the strongest possible terms that these monitors were abducted and have remained as hostages for so long. We call on all OSCE participating States, including Russia, to use their influence to secure the immediate and unconditional release of these monitors, to guarantee the security of the OSCE monitoring teams as they carry out their mandate throughout Ukraine, and to denounce the use of OSCE monitors or observers as hostages.
This past week we witnessed the inauguration of Petro Poroshenko as President of Ukraine. He was elected following credible, well-run elections praised by international observers, including ODIHR and the OSCE Parliamentary Assembly. President Poroshenko is the first president of Ukraine to win in a first round vote, having collected a majority of votes cast across the country, having come in first in every oblast. His inauguration represents a unique opportunity to make progress toward a peaceful resolution of the crisis in Ukraine. Unfortunately, Russia so far has failed to recognize Poroshenko as Ukraine's president and has continued destabilizing actions in eastern Ukraine, including support for large-scale attacks on Ukrainian government positions, border posts, and convoys in Donetsk and Luhansk regions. Furthermore, it is evident that heavily armed and organized fighters from Russia, particularly from the regions of Chechnya and North Ossetia, are playing an overt role in the violence. Today, news reports contain acknowledgment by Russia of material support being channeled to separatists, and news of a visit by the so-called leader of the illegal Donetsk People's Republic, [Denis] Pushilin, to Moscow. This is not a revelation, but exposure of yet another lie to be added to the tally. Such actions subvert every good faith effort at dialogue and call into question the seriousness with which the Russian Federation engages in diplomatic discussions.
Former Russian Government Advisor: Putin's Actions In Ukraine Aren't a New Cold War – They're a Real War
In our Windows on Eurasia
column, Paul Goble provides a summary of the widely-discussed address
of economist and former Russian government advisor Andrey Illarionov at
NATO's Palriamentary Assembly May 31st. In a speech titled "The Fourth
World War" among the best of his career, Illarionov spells out in 10
theses why Putin's military action in Ukraine is a real war, not a
return to the Cold War, bearing more resemblance to Soviet invasion of
the Baltic states or even the 1917 Bolshevik coup; it is not a
disruption of good relations which were not as imagined but the
continuation of Moscow's foreign aggression begun in August 2008 with
the Georgian war.
Putin's purpose is to divide Europe and the US
over relations with Russia but should be met with a concerted long-term
response, with a particular focus on counter-propaganda.
Read Illarionov's Ten Theses on Putin's Aggression in Ukraine
fetching tweet…
Humanitarian Crisis Growing In Slavyansk
The situation facing civilians who still remain in Slavyansk, effectively the key stronghold of the Russian-backed separatist gunmen, continues to decay. Alec Luhn of The Guardian reports from the city:
Most residents in the besieged city have been without water, electricity and gas for the past week. Food supplies are limited, and grocery stores smell of rotting food from the lack of refrigeration. Dozens of people queue for drinking water.
About 270 people have died in the east of the country since Kiev launched its "anti-terrorist operation" two months ago, Ukraine's health ministry said on Wednesday. Of those killed, 225 were in Donetsk region, including Slavyansk, where fighting has been heaviest. At least two children have died of shrapnel wounds in Slavyansk this month, according to the health ministry. However, past government estimates have been low, excluding deaths in rebel-held territory.
Appalling conditions in rebel-held towns have caused thousands to flee. The exodus from Slavyansk gathered pace when Ukrainian army shelling intensified at the end of May, with most residents going to the nearby city of Svyatogorsk were they are dependent on the goodwill of locals for housing and food. About 15,000 to 20,000 refugees from Slavyansk have arrived in the city since the end of May, according to mayor Alexander Dzyuba.
NATO Calls Reports Of Russian Tanks In Ukraine a 'Serious Escalation'
This tweet says it all:
The full statement from NATO Secretary General Anders Fogh Rasmussen:
I am concerned about reports that pro-Russian armed gangs are acquiring heavy weapons from Russia, including Russian tanks. We have seen reports that Russian tanks and other armoured vehicles may have crossed the border into eastern Ukraine. If these reports are confirmed, this would mark a serious escalation of the crisis in eastern Ukraine.
I continue to urge Russia to complete the withdrawal of its military forces on the border with Ukraine, to stop the flow of weapons and fighters across the border, and to exercise its influence among armed separatists to lay down their weapons and renounce violence. We call on the Russian Federation to meet its Geneva commitments and cooperate with the government of Ukraine as it implements its plans for promoting peace, unity and reform.
The picture in that tweet shows a tank that Ukraine says crossed from Russia into Ukraine yesterday. There are three key questions:
1) Where did the separatists get the tanks?
2) Are they captured Ukrainian tanks, Russian tanks, or tanks from some other source?
3) How did the tanks get to where they are now? Did they pass through Russia or get there with the help of the Russian military?
Yesterday we posted a story that three T-64Bs were captured by Russian-backed separatists near Lugansk. But those vehicles were reportedly captured on or before June 9th. Also, to get from Lugansk to where they were spotted they would have to travel through the military front of Ukraine's anti-terror operation — unless they traveled north first and traveled through Russia to loop around and re-enter Ukraine west of Rostov-On-Don.
But where did the tanks originally come from? The report we just mentioned was only posted on a single website that we know of, and it is Russian. Another theory was that the the T-64Bs were captured in Crimea. But Ukraine's military has had checkpoints at the bridges that connect Crimea to mainland Ukraine since long before the annexation. If the tanks are from Crimea, they would have had to be shipped, likely by Russian military transport, to Ukraine (or to Russia and then on to Ukraine). But there's no evidence (yet) to suggest that this happened.
Throwing a wrench in the works, the Ukrainian military expert Dmitry Tymchuk says all Ukrainian tanks are accounted for but Russia has T-64s in storage (translated by The Interpreter):
According to operational information from Information Resistance, the tanks abandoned by the Donbass terrorists yesterday from Russia were T-64BV tanks (earlier we had indicated, based on preliminary information that these were T-72 tanks, however according to more accurate data these are T-64s.)
Such tanks were in the arsenal of the coastal guard brigades of the Ukrainian Navy in the Crimea. However, according to our information, all Ukrainian T-64BV tanks (41 units) were withdrawn to the mainland of Ukraine.
In addition, according to information from international organizations, at this time, there are T-64BV tanks in Russia. In particular, they were withdrawn from combat use by the Russian army, but are located in the RF Defense Ministry's warehouses.
In 2013, there were about 2,000 T-64 tanks with various modifications including the T-64BV in vehicle and armaments warehouses in Russia (according to the International Institute for Strategic Studies, see The Military Balance 2013).
We have not confirmed that Russia has T-64s in storage. The most likely answer may still be the first one — that these tanks were captured in a recent raid by separatists. But as we've pointed out, that still doesn't explain how they got into southeastern Ukraine without a lot of people noticing.
A New Offer On The Price of Ukraine's Gas
There may be some progress in the negotiations over Ukraine's natural gas supply. RFE/RL reports:
The head of Ukraine's gas company has said that Kyiv may pay some of its overdue payment to Gazprom on June 13 if Russia agrees to charge $326 per 1,000 cubic meters of gas for 18 months.
Naftohaz chief Andrei Kobolev told reporters in Kyiv today that his corporation was "prepared to pay $1.9 billion if we make the deal."
Kobolev said the price of $326 per 1,000 cubic meters — higher than what Kyiv wants to pay and lower than what Russia demands — was proposed by the European Commission during recent talks.
The problem is that we've seen this play before. So far Russia has been unwilling to come down on prices, and Ukraine has been unwilling to pay its debts until that happens. The fact that Kobolev is proposing a compromise, and the fact that he says this is a proposal that was discussed during recent talks, could indicate that progress is being made toward an ultimate solution.
Separatist Fighters Reportedly Fire Grad Rockets And Kill Civilian
Novosti Donbassa reports that separatist fighters have accidentally set fire to their own Grad rocket launcher and destroyed the depot of a vegetable company by mistake in the Donetsk town of Dobropol.
A comment on a local VKontakte group read (translated by The Interpreter):
"From this area they fired several volleys from both installations towards the town of Dobropol. While firing a salvo, one of the launchers caught fire. Not all the ammunition was fired. Out of the 40 rockets, 14 remain unused, and one of them malfunctioned and fell on a barn near the launch site. They allegedly fired at a Ukrainian checkpoint which is on the outskirts of the town, towards Svyatogorsk. But a mistake was made in the firing adjustments and all of the missiles fell on the Perspektivy vegetable depot, where a woman who worked for the company was seriously injured. Two rockets fell on a neighbouring premises, where a young man who was working as security guard died."
The Novosti Donbassa report says the area is now cordoned off by the Ukrainian military as the separatists had placed mines around the burnt launcher.
This video shows damage following the firing:
Here are photos from the Novosti Donbassa report of the launcher and the aftermath of the incident, including visible impact craters and the dead security guard.
Video Of Ukraine Military's Assault on Mariupol
Watch this video carefully. Then try to remember that the rag-tag group of men, fighting in mismatching uniforms and a variety of weapons and using what looks like a garbage truck to assault the enemy are actually Ukrainian soldiers, not separatists.
The Ukrainian military did, however, take over the town and disable separatist armored personnel carriers in the process of the assault. These are likely either National Guard troops or part of the Azov battalion, as Ukrainian regular forces were also engaged in the fight.
Reports That Separatist Tanks Either Broken Down or Destroyed
Ukrainska Pravda reports that Anton Geraschenko, an adviser at the Ministry of Internal Affairs, announced at a press conference (translated by The Interpreter):
"These tanks are now near the town of Chasov Yar in the Artyomovsk district of the Donetsk oblast. The are already in a practically un-usable condition. According to our data, these tanks are moving no further because they have broken down."
The report says that Geraschenko noted that the tanks had likely been midhandled due to the inexperience of their non-professional milita crews.
UNIAN reported that Geraschenko then warned the separatists to abandon the tanks for the sake of their lives as they would soon be destroyed from the air.
According to the MVD the tanks (identified as T-64s) had belonged to the Ukrainian Armed Forces but had been seized during the Russian invasion of Crimea. They were reportedly shipped by sea to the Russian mainland.
Geraschenko said that by using captured Ukrainian tanks, Vladimir Putin was attempting to manipulate international opinion by being able to point out that Russian tanks had not entered Ukrainian territory.
The time of the press conference is not given in either report (both filed at around 10 a.m. GMT). But at 6:22 GMT Dmytro Tymchuk of Information Resistance claimed that 2 of the tanks had been destroyed by Ukrainian forces.
Another video of the tanks convoy in the town of Torez is below:
Major Government Operation in Mariupol
The minister of the interior, Arseniy Avakov, announced on his Facebook page this morning that the " active phase of the ATO" had begun in Mariupol.
He writes (translated by The Interpreter):
A planned operation is under way to eliminate the fortified terrorist area headed by the crime boss who goes by the name 'The Chechen'. The area of operations in the centre of Mariupol has been cordoned off.
Units of the Ukrainian National Guard and the Azov and Dnipr-1 special operations units of the Interior Ministry are participating in the operation. Checkpoints around Mariupol are supporting the armed forces.
The operation began at 4:50 a.m. [1:30 GMT]. It is going according to plan. Successfully. All of the key terrorist points of control have, at this time, been brought under control. A BRDM has been knocked out and terrorist sniper positions have been destroyed. Losses on the terrorists' side are high. There are no losses on ours – two soldiers have been wounded.
The operation is continuing.
It is a shame that Mariupol was woken up this morning by gunfire – but the city of the steelworkers must not live and work under the dictatorship of bandits!
At 10:08 (7:08 GMT) Avakov added an update:
11 terrorists have been detained, two explosive devices have been neutralised. A car was seized at a checkpoint trying to break through in the direction of Donetsk. The car was fully loaded with weapons. The situation in the city is stable.
And at 10:34 (7:34 GMT):
The Ukrainian flag is flying over the Mariupol City Council!
Mariupol-based news site 0629.com.ua posted this photo of National Guard forces on Lenin Street in the city:
This tweet from the Euromaidan Twitter account appears to show the knocked out separatist BRDM armoured reconnaissance car that Avakov described:
Translation: Morning in Mariupol. Glory to Ukraine.
Here are some more photos from 0629 of the events today:
ATO Borders Denis Pushilin Donbass Donetsk People's Republic Lugansk Mariupol Russian invasion SBGS separatists Slavyansk tanks Ukraine
2 Russia-Backed Fighters Killed; Unrecognized Election Campaign Begins in Self-Proclaimed DNR
10 Russia-Backed Separatist Fighters Reported Killed in Mine Blast, 2 in Battle
Two Ukrainian Soldiers Wounded; 1 Civilian Wounded; DNR Drone with Explosives Crashes
9 Policemen Injured in Mariupol-Kiev Soccer Match
1 Ukrainian Soldier WIA; Volker Says Russian Peacekeeping Plan Would 'Deepen Division' Without Monitors' Access to Border
2 Ukrainian Soldiers Wounded in Action; Russian Helicopter Violates Ukrainian Air Space


text:
hospodars
plural form of hospodar
"Hospodars." YourDictionary. LoveToKnow. www.yourdictionary.com/hospodars.
Hospodars. (n.d.). In YourDictionary. Retrieved from https://www.yourdictionary.com/hospodars
During the campaign Peter had entered into alliance with the hospodars of Moldavia and Walachia, respectively Demetrius Cantemir and Constantine Brancovano, from whom he had received material assistance.
Russia, desirous of deriving some return for the support which she had given the sultan during his rupture with the French, induced the Porte to address to her a note in which the right of intervention in the affairs of the principalities, conferred on her by the treaty of Kainarji and reaffirmed in the convention of Ainali Ka y ak, was converted into a specific stipulation that the hospodars should be appointed in future for seven years and should not be dismissed without the concurrence of the Russian ambassador at Constantinople.
Its terms were: the confirmation of the Treaty of Bucharest and the opening of the navigation of the Black Sea to the Russian flag; a stipulation that the hospodars of Walachia and Moldavia should be elected by the boyars for seven years, their election being confirmed by the Porte which, however, had no power to dismiss them without the concurrence of the Russian ambassador at Constantinople; finally, Servia's autonomy was recognized, and, save in the fortresses, no Mussulman might reside there.
Taking a leading part in the political movements of the time, he came into conflict with the newly appointed Greek hospodars, and was exiled to Rumelia.
These armaments, which cost Matthias 1,000,000 florins per annum, equivalent to 200,00O, did not include the auxiliary troops of the hospodars of Walachia and Moldavia, or the feudal levies of the barons and prelates.
hospodar
WORDS NEAR hospodars IN THE DICTIONARY
hospiticides
hospitium
hospitiums
hoss-opera
hosses


Mustafa Center is looking for talented and skilled people in our community at all times.
More details will be posted very soon. In the meantime, if you would like to volunteer, please email contact@mustafacenter.org.
Mustafa Center is run and is able to serve you solely based on your generous donations. All your donations are tax deductible.


text:
Pagiphora is a genus of cicadas in the family Cicadidae, found in the western Palearctic. There are about five described species in Pagiphora.

Pagiphora was formerly placed in the tribe Cicadettini, but is now the sole genus of the tribe Pagiphorini.

Species
These five species belong to the genus Pagiphora:
 Pagiphora annulata (Brullé, 1832)
 Pagiphora aschei Kartal, 1978
 Pagiphora hauptosa Boulard, 1981
 Pagiphora maghrebensis Boulard, 1981
 Pagiphora yanni Boulard, 1992

References

Further reading

Cicadettinae
Cicadidae genera

redpajama_set_name: RedPajamaWikipedia


text:
Rep. Harris Hopes GAO Will Examine Md. Health Exchange
Filed Under:Government Accountability Office, Health Exchange, Healthcare, Rep. Andy Harris
ANNAPOLIS, Md. (AP) — A Republican Maryland congressman says he believes the investigative arm of Congress should look into problems with the state's health care exchange website.
The U.S. Government Accountability Office on Wednesday agreed to a request from a group of House Republicans to audit how $304 million in federal grants were spent on the Oregon health exchange website. A GAO spokesman says other states with troubled exchanges will be included as the investigation goes forward.
Rep. Andy Harris, Maryland's only Republican congressman, has asked the inspector general of the federal Health and Human Services Department to investigate.
Chris Meekins, a spokesman for Harris, says the congressman still believes the HHS inspector general also should look into problems with the Maryland health exchange, which has been plagued by computer problems.
(Copyright 2013 by The Associated Press. All Rights Reserved.)


text:
A Trans Man Steps Into The Ring – And Wins His Debut As A Professional Boxer
By Laurel Wamsley
Published December 13, 2018 at 5:26 PM CST
Patricio Manuel, the first openly transgender man to box professionally in the U.S., faced off against Hugo Aguilar on Saturday evening at a casino in Indio, Calif. The judges declared Manuel the winner.
If you just happened to be in the crowd at a super featherweight bout in Indio, Calif., on Saturday evening, you might not understand the importance of that particular boxing match.
But for Patricio Manuel, it represented 12 minutes of triumph. With that fight, Manuel, 33, became the first openly trans man to compete in a professional boxing match in the U.S.
It was moment of joy following a long journey after Manuel first stepped into the ring as a high schooler in Gardena, a small city south of Los Angeles.
"The first week I fell in love with the sport and never looked back," Manuel tells NPR's Steve Inskeep. He was drawn by boxing's rigor — "the specific, very grueling training from boxing, as well as the opportunity to compete consistently is what really attracted me to the sport."
Boxing entered his life at the same time as another fateful event for Manuel: puberty.
"I had always seen myself as a boy, even though society basically kept telling me no," he says. "And when I started going through puberty, it was like nature being like, no — you actually are a girl, no matter how much you don't want to be."
Manuel trained hard, developing a high-pressure style, throwing lots of shots at his opponents. He worked up the ranks of women's boxing, winning five amateur championships – and qualified for the 2012 Olympic trials, the first time women's boxing would be in the games. But after sustaining an injury in the first round at the trials, Manuel had to withdraw.
It was a career setback, but it forced a personal reckoning.
"Boxing was this thing I love, but it was also a distraction from me really looking at myself, and being like 'Who are you, really? What will make you happy?' " he says. "I was just always like, 'Boxing makes me happy.' But there's more to me than just a sport. And when the sport was taken away from me, I really had to look at myself, and be like, 'There's more to this than just losing the fight.'"
Manuel realized he had been lying to himself about being known as a female athlete and competing in women's boxing: "I really was a man that wanted to be in boxing, but I was afraid that I would lose my ability to compete."
It was time. Manuel decided to go by the name Pat, to use he/him/his pronouns and live publicly as a man. He also began taking hormones and medically transitioning, having a mastectomy and then surgery to give him a male-shaped chest.
But other complications arose now. His longtime gym told him that he could train there, but only if he didn't tell anyone — an impossibility for Manuel. "I've never lived my life in a closet, and I refuse to compromise that way. So I walked out of that gym and I never came back."
He found a new gym that welcomed him, and he fought two amateur bouts in 2016 – winning one, losing the other. But it was hard to find opponents, and he had to get used to taking on boxers who fought in a style more like his own.
"A lot of the male fighters — especially in Los Angeles, which is a primarily Mexican-populated boxing area — a lot of it's hunting someone down and ripping them to the body, which used to be my style in the amateurs," Manuel says. "But I faced a lot of female fighters who were really excellent boxers who worked on getting their distance, getting their points, moving out of the way. So it was an adjustment to turn from being the person who was always chasing down and breaking down the body of other fighters, to having someone do that to me."
But he kept up his training regimen, and with some new support from Golden Boy, Oscar de la Hoya's boxing promotion company, Manuel was able to get his pro license to box as a man. Golden Boy also found him an opponent.
So on Saturday evening, in an eight-fight card at a casino in Indio, Manuel battled a man named Hugo Aguilar. After four rounds, the judges declared Manuel the winner.
As Manuel was interviewed in the ring about what it all meant, boos and whistles could be heard from the crowd. But Manuel wasn't shaken, and he says he wasn't angry, either.
"I'm a black trans man. I've had people saying cruel, hateful things to me my entire life. People booing me – it's more about them than me. They don't know me. They don't know what I've been through. They don't know how much I love this sport, how happy I was in that moment. I refuse to give them power over me by feeling even angry toward it."
And, he says, he'll be back.
"This wasn't a one-show, this wasn't a publicity stunt. This is something I love, something I've invested my entire life to, this is something I've sacrificed for. This is just the start."
NPR's Reena Advani, Merrit Kennedy and Kevin Tidmarsh contributed to this story.
Laurel Wamsley is a reporter for NPR's News Desk. She reports breaking news for NPR's digital coverage, newscasts, and news magazines, as well as occasional features. She was also the lead reporter for NPR's coverage of the 2019 Women's World Cup in France.
See stories by Laurel Wamsley


Since the beginning we have held the theatrical training of adults, teenagers, and children and people with various disorders as a priority.
We bring this activity forward with courses for Higher Theatre Education, organisation of workshops in schools of various order, and degrees and seminars.
The objective of our educational activities is the dissemination of theatre making and of its knowledge, starting from the idea of a theatre workshop in which one can gain the tools to bring to life the expressive potentials of themselves.
In our vision, theatre is the place of exchange and confrontation where you can meet different winds and different and ways of working.
Our courses originate from the conviction that every one of the participants can be a creative wind that both enriches and is enriched by encounters with others.
The Centre for Theatrical Development is realised within the project Urban THEATRE EXPERIENCE supported by FUNDER35 – fund for youth cultural enterprise.
Since the beginning we have prioritised the development of the theatre field aimed at adults, teenagers, children, and people with various disabilities.
We bring this activity forward through the courses of the Centre for Theatrical Development, the organisation of workshops in schools of different grade, and the planning of seminars.


Buika is a promising artist who has been voted among the "50 Great Voices" of the world by NPR. Coming from MaJorca Island, Buika has emerged as a real innovator. For jazz enthusiasts, she is a must-see musician. Her unmistakable African roots along with her cosmopolitanism, developed in clubs in Spain and Americanized all through surreal period in Vegas, enlightens the exceptional self-possession of Buika. With your Buika tickets you will experience watching a unique artist playing live.
Buika's exclusive singing and distinctive style makes her a wonderful and rare artist to listen to. Her fame is rapidly growing in Europe and in United States. Buika is quickly being recognized and seen among the most dynamic and individualistic artists in modern vocal jazz. Buika's collaboration with Chucho Valdes in 2009 won her a Grammy Award in the category of "Best Traditional Tropical Album." The next year, Buika released a duet "You Got Me" with Seal. Buika also made a guest appearance as a vocalist on the Traveller CD of Anoushka Shankar. Buika has released her album "En Mi Piel" that is an exhaustive compilation. This two-disc release showcases the breathtaking range and diversity of Buika and her progress as an artist. The record features signature numbers from her previous releases as well. Buika cites Michael Jackson and Whitney Houston as her idols.
With every new endeavor, the fan base of Buika seems to grow. Buika is a keen performer and has won millions of her fans' hearts with her exciting performances. Buika tickets are always high in demand and the live concert audiences of Buika include some great luminaries such as Alicia Keys, Antony Hegarty, Alejandro Sanz and Gloria Estefan. Buika makes her audiences enjoy her music, transporting them to a place where music reigns. Buika thrillingly celebrates her passion for music. The discography of Buika includes Nina de Fueg, El Ultimo Trago, The Skin I Live In and En Mi Piel among others. Buika is a highly praised flamenco synthesis vocalist with a breakthrough release to her credit that sold more than 100,000 units in Spain. Critics have described Buika's voice as "achingly beautiful", "sultry" and "sonorous".
Buika is known to hook young audiences with her originality and the unique blending of other influences. The fusion of soul, flamenco with jazz and dance rhythms in the music of Buika reflects her own life story. Buika sings with emotions of loneliness as she states, "I sing against emotional dictatorships". She is one among the most spontaneous and lively artists in the existing scene of Spanish music. Buika manages all types of rhythms and she sure knows how to put on a great show. She is known as one of those artists who can captivate the audiences with her brilliant voice, talent and high energy on stage. Buika's sensuality of voice makes her a great artist to enjoy live.
Not only is Buika a soulful and gifted artist but she is also a down to earth performer. Fans call her an amazing woman with powerful stage performance and a voice that is beyond words. Her vocals are majestic and incredible. Buika tickets are must to have in order to watch and listen to this amazing singer live. When attending her live concert, you can see the great passion and love for music in her voice. She also makes the audience connect with her music. You will be enthralled to see her once and would want to come see her over and over again. It is going to be a remarkable event and a lifetime opportunity to watch Buika performing on stage. The audiences will find this show one of the most enjoyable events ever, so make sure you are there. Have a great time!
As of 22/04/2019 the average Buika tickets will cost you $395 for the event being held on 09/05/2020 at Carnegie Hall - Isaac Stern Auditorium, New York. The minimum get in price is $62 at Carnegie Hall - Isaac Stern Auditorium.


text:
LISTEN: First New 'Sounds' From The Normal In 40 Years
Christian Eede , July 22nd, 2019 15:09
Daniel Miller's tribute to John Cage's '4'33"' features on a new Mute box set
With STUMM433, the Mute box set which forms part of the label's 40-year anniversary celebrations in 2019, set to arrive post-summer, the label has now shared label boss Daniel Miller's contribution to the release under his The Normal guise. You can hear it above.
The box set sees more than 50 Mute artists - past, present and future - all offer up their own interpretation of John Cage's famed composition '4'33"', and Miller's is the second of those to be unveiled by the label following on from Laibach's effort. The track marks Miller's first work as The Normal since the very first Mute release in 1978, 7" single T.V.O.D. / Warm Leatherette.
Miller apparently recorded his tribute to Cage's composition at the North London site of Mute's original offices: 16 Decoy Avenue. "John Cage's '4'33"' has been present in my musical life for as long as I can remember as an important and inspiring composition," Miller says. "When the idea of every Mute artist doing their own interpretation of the piece came up during a conversation with Simon Fisher Turner, I immediately thought this was the perfect way to mark the label's MUTE 4.0 (1978 > TOMORROW) series."
Amongst the other acts to have contributed their own interpretations of '4'33"' are Goldfrapp, Depeche Mode, Erasure, Liars, Yann Tiersen and Irmin Schmidt. You can find out more about the release, which is due out on October 4, here.
Daniel Miller Interview: Mute Records & The Synth Britannia Revolution »
Daniel Miller Discusses Mute's Introduction Series »
'A Primal Bang & A Shout': An Interview With S.C.U.M »
Mute Talk Liberation Technologies Imprint »
How Soon Is Now? Extract: The Rise Of Blast First »


More here and also here.
What did he Call Limbaugh, Beck et al?
…small men behind the curtain.
and what's Limbaugh's place in the world according to David Brooks?
In the media world, he is a giant. In the real world, he's not.
It is difficult to think of hypocrisy more abhorrent.


text:
Monster Energy Supercross—The Official Videogame Released
February 13, 2018 12:05pm | by: Press Release
MILAN, Italy — The time has come: Monster Energy Supercross—The Official Videogame is now available in stores and on digital for PlayStation 4, Xbox One, Windows PC/STEAM, and Nintendo Switch.
Starting from today, users can purchase also the Monster Energy Supercross Season Pass for $14.99 to get additional exclusive content. The Season Pass, which is available for PlayStation 4 and Xbox One, includes five DLCs in total, three of them available at launch:
A DLC with 60 liveries—10 for each manufacturer —and 10 rider tracksuits to customize bikes and riders
An exclusive set of 15 additional icons and 15 butt patches to further expand your customization options
The Credit Multiplier DLC
Shortly after, users will get access also to two additional DLCs:
Monster Energy Cup, set to offer all fans the exact replica of the 2017 Monster Energy Cup track at Sam Boyd Stadium in Las Vegas for both offline and online modes
Compound, a training space with two new tracks for players to train on supercross and motocross circuits.
Featuring innovative track designs and set inside recognizable and prestigious stadiums, Monster Energy Supercross—The Official Videogame is the most realistic and immersive supercross fan experience.
The official 2017 Monster Energy Supercross Season
Official bikes, iconic tracks, including Daytona, and athletes from both 250SX and 450SX Classes: Chad Reed, Ken Roczen, reigning champion Ryan Dungey, and others will race across official challenging tracks
An exciting career mode with extreme customization options for customizing both riders and bikes with the maximum flexibility to advance in racing classes
A powerful track editor for endless gameplay possibilities. Thanks to the track editor, players can create their own custom tracks, race on them, and share them online. Once uploaded on the server, users can "like" other gamers' creations, vote on their favorite tracks, and increase the prestige of the track designer.
Top-notch graphics for a world-class game quality and a next-level immersive experience
Online modes, with multiplayer for up to 12 players who can ride on both official and custom tracks
The track editor and online modes will only be available on PlayStation 4, Xbox One and Windows PC/STEAM.
Standard Edition of Monster Energy Supercross—The Official Videogame is available for PlayStation 4 and Xbox One for $59.99, while the Special Edition, including Full Game and Season Pass, is $69.99. The game for Windows PC/STEAM is available at $49.99 and for Nintendo Switch at $59.99.
The Supercross Championship also released on mobile platforms, with the app Monster Energy Supercross Game now available on iOS.
Website: www.supercrossthegame.com
Facebook: facebook.com/SupercrossTheGame
Instagram: Instagram.com/supercrossthegame
Hashtag: #supercrossthegame
Fredrik Noren Out for Remainder of 2018 8:35am
Justin Bogle Undergoes Successful Surgery; Malcolm Stewart Staying with Team 3:00pm


text:
Stealthy startup Human debuts headphones shaped like ears, calls them an 'evolution of personal audio'
by Taylor Soper on September 20, 2016 at 5:01 am September 20, 2016 at 6:38 am
Human Inc. today opened pre-orders and unveiled more details about its new headphones and accompanying platform that it describes as an "evolution of personal audio."
The stealthy Seattle startup, which GeekWire first discovered after it raised a $700,000 seed round in December, on Tuesday launched a crowdfunding campaign on Indiegogo for "Sound."
The wireless headphones are shaped like ears and "act as an extension of the human body," as Human describes them.
There are no buttons — users control music and answer phone calls by swiping gestures on the headphones themselves. The device, which holds 12 hours of battery life and comes with an induction charging station, turns off when the earphones magnetically come together, and turn on when they come apart.
The headphones also let users share audio from the same device with up to 20 individuals; block out or let in more noise in one's surrounding environment; and can turn into a loudspeaker by placing the earphones together.
"The future of audio is more than wireless headphones," Human co-founder Joe Dieter said in a statement. "The industry has been playing it safe and only inching forward its consumer audio offerings."
There is a one-day pre-order sale today for $170 per unit; the headphones will start at $200 during the crowdfunding campaign. They retail at $400 per unit.
Wireless headphones have been around for a while, but Human's timing is notable, given that Apple's newest iPhone will no longer feature a headphone jack and may convince more people to buy wireless headphones.
Perhaps even more intriguing is the company's plan for additional features. It is offering a limited number of pre-sale customers the chance to beta test technology inside the device that provides instant language translation, advanced biometric monitoring, and sleep tracking.
Human co-founders Benjamin Willis and Joe Dieter.
Dieter founded the 40-person company last year with CEO Benjamin Willis. Bill Moore, previously CEO of companies like RootMetrics and Allrecipes, became Human's president and executive chairman earlier this year.
Human raised a $5 million investment round this past April, pushing total funding to date is near $6 million. Investors include folks like Fred Warren, former managing director of Brentwood Associates; Darling Ventures Managing Director Daniel Darling; and Sugar Mountain Capital CEO Kurt Beecher Dammeier.
[Editor's Note: Human is a GeekWire advertiser.]
Car2go launches airport car-sharing service in Seattle, joins BMW and Zipcar
Surprise: Microsoft unveils new Nokia cell phone after agreeing to sell the business
Filed Under: Startups Tagged With: audio • headphones • human
Microsoft invests in $21M round for wireless over-the-ear headphones startup Human
Hands-on with the $350 Surface Headphones: Why is Microsoft making them, and are they any good?
Wireless headphone startup Human raises $21.2M, product still in development 4 years after launch
Listen to this: Microsoft introduces $349 Surface Headphones, a new product category available this fall


text:
\section{How to Use this Template}

\section{\label{chapter:introduction}Introduction.} In this study our aim is to develop the possibilities of numerically solving variational problems that appear in epidemic dynamics models. The algorithm that we will use for this purpose will be a modified Monte Carlo method. In a few sentences, we recall those stages of the Monte Carlo method development and its improvement, which constitute the essence of the applied approach (cf. \citeauthor{Martino2018}~\cite[in sec. 1.1]{Martino2018}).

Before constructing the computer, the numerical analysis of deterministic models used approximations that aimed, among other things, at reducing and simplifying the calculations. The ability to perform a significantly larger number of operations in a short time led Ulam (cf. \citeauthor{Met1987:BeginMC}~\cite{Met1987:BeginMC}) to the idea of transforming a deterministic task to an appropriate stochastic task and applying the numerical analysis using the simulation of random quantities of the equivalent model (cf.~\citeauthor{MetUla1949:MCM}~\cite{Met1987:BeginMC},). The problem Ulam was working on was part of a project headed by von Neumann, who accepted the idea, and Metropolis gave it the name of the Monte Carlo Method \cite[in sec. 1.1]{Martino2018}.

The resulting idea allowed us to free oneself from difficult calculations, replacing them with a large number of easier calculations. The problem was the errors that could not be solved by increasing the number of iterations. Work on improving the method consisted in reducing the variance of simulated samples, which led to the development of the Importance Sampling (IS) method in 1956 (cf. \citeauthor{Mar1956:IS}~\cite{Mar1956:IS} ).

Computational algorithms in this direction are a rapidly growing field. Currently, the computing power of computers is already large enough, to provide an opportunity to solve problems that can't be solved analytically. However, some problems and methods still require a lot of power. A lot of research is devoted to finding or improving such methods. The development of this field is currently very important. There are many models describing current problems. Not all offer the possibility of an analytical solution. That is why various studies are appearing to find best methods for accurate results. the Cross-Entropy method helps to realize \textbf{IS} and it may, among other things, be an alternative to current methods in problems in epidemiological models.

\subsection{Sequential Monte Carlo methods.}
The \textbf{CE} method was proposed as an algorithm for estimation of  probabilities of rare events.  Modifications of cross entropy to minimize variance were used for this by \citeauthor{rubinstein1997optimization} in his seminal paper \cite{rubinstein1997optimization} in \citeyear{rubinstein1997optimization}. This is done by translating the "deterministic" optimization problem in the related "stochastic" optimization and then simulating a rare event. However, to determine this low probability well, the system would need a large sample and a long time to simulate.

\subsection{Importance sampling.}
Therefore, it was decided to apply the importance sampling (\textbf{IS})\label{IS::important}, which is a technique used for reducing the variance. The system is simulated using other parameter sets (more precisely, a different probability distribution) that helps increase the likelihood of this occurrence. Optimal parameters, however, are very often difficult to obtain. This is where the important advantage of the \textbf{CE} method comes in handy, which is a simple procedure for estimating these optimal parameters using an advanced simulation theory. \citeauthor{sani2007optimal} \cite{sani2007optimal} have used the Cross-Entropy method as the support of the importance sampling to solve the problem of the optimal epidemic intervention of HIV spread. The idea of his paper will be adopted to treat the \textbf{SIRC} models, which has a wide application to modeling the division of population to four groups of society members based on the resistance to infections (v. modeling of bacterial Meningitis transmission dynamics analyzed by \citeauthor{AsoNya2018:Meningitis}~\cite{AsoNya2018:Meningitis}, \citeauthor{Ver2008:SIRC}~\cite{Ver2008:SIRC}).   The result will be compared with an analytical solution of \citeauthor{casagrandi2006sirc}~\cite{casagrandi2006sirc}.

\subsection{\label{section:cemethod}Cross-Entropy method.} This part will contain information on the methods used in the paper. The Cross-Entropy method developed in \cite{rubinstein1997optimization} is one of the versions of the Monte Carlo method developed for problems requiring the estimation of events with low probabilities. It is an alternative approach to combinatorial and continuous, multi-step, optimization tasks. In the field of rare event simulation, this method is used in conjunction with weighted sampling, a known technique of variance reduction, in which the system is simulated as part of another set of parameters, called reference parameters, to increase the likelihood of a rare event. The advantage of the relative entropy method is that it provides a simple and fast procedure for estimating optimal reference parameters in IS.

Cross entropy is the term from the information theory (v. \cite{Par1969:Entropy}, \cite[Chapter 2]{CovTho2006:ElementIT}). It is a consequence of measuring the distance of the random variables based on the information provided by their observation. The relative entropy, the close idea to the cross entropy,  was first defined by  Kullback  and  Leibler~\cite{KulLei1951:Information} as a measure of the distance  $\mathcal{D}(p||q)=\textbf{E}_p\log\frac{p(X)}{q(X)}$ (v. \cite[(1.6) on p. 9; Definition on Sec. 2.3  p. 19]{CovTho2006:ElementIT}, \cite{Ing2014:TIandSM}) for two distributions $p$, $q$  and is known also as Kullback-Leibler distance. This idea was studied in detail by Csisz\'ar \cite{Csi1967:Information} and Amari~\cite{Ama1985:Diff}. The application of this distance measure in the Monte Carlo refinements is related to a realization of the importance sampling technique.

The \textbf{CE} algorithm can be seen as a self-learning code covering the following two iterative phases:
	\begin{enumerate}\itemsep1pt \parskip1pt \parsep1pt
		\item Generating random data samples (trajectories, vectors, etc.) according to a specific random mechanism.
		\item Updating parameters of the random mechanism based on data to obtain a "better" sample in the next iteration.
	\end{enumerate}
Now the main general algorithm behind the Cross-Entropy method will be presented on the examples.

\subsection{Application to optimization problems.} This section is intended to show how extensive the \textbf{CE} method is. The examples will show the transformation of the optimization problems to the Monte Carlo task of estimation for some expected values. The equivalent \textbf{MC} task uses the \textbf{CE}  method. Other very good examples can be found in \cite{Rub1999:CEMethod}.

The first variational problem presented in this context is the specific multiple stopping model. Consider the so-called secretary problem (v. Ferguson~\cite{Fer1989:Who} or the second author's paper~\cite{sza1982:Ath}) for multiple choice, i.e. the issue of selecting the best proposals from a finite set with at most $k$ attempts. There is a set with $N$ objects numbered from $1$ to $N$. By convention, the object with the number 1 is classified as the best and the object with the number $N$ as the worst. Objects arrive one at a time in a random order. We can accept this object or reject it. However, if we reject an object, we cannot return. The task is to find such an optimal stopping rule that the sum of the ranks of all selected objects is the lowest (their value is then the highest). So we strive for a designation
	\[
	\inf_{\tau}\mathbf E X_{\tau}=\mathbf E X_{\tau^{*}}.
	\]
Let $a_{i}$ be the rank of the selected object. The goal is to find a routine for which the value of $\mathbf E (a_{\tau_{1}}+\dots+a_{\tau_{k}})$ for $k\geqslant2$, is the smallest. More details are moved to~\ref{SumOfRank}.

Next, the problem was transformed to the minimization of the mean of sum of ranks. The details are presented in the section~\ref{MinMeanSumRank}.  This problem and its solution by \textbf{CE} was described by \citeauthor{Pol2010:CEBCP}~\cite{Pol2010:CEBCP}. Its correctness was checked and programmed in a different environment by \citeauthor{Sta2019} in \cite{Sta2019}.

The second example is presented in details in Section~\ref{VehRP}. It is a formulation of the vehicle routing problem (v. \cite{Dro2002:VRSD}, \cite{CheHom2005:vehicle}). This is an example of stochastic optimization where the Cross-Entropy is used.

\subsection{Goal and organization of this paper. }
In the following parts of the paper, we will focus on models of the dynamics of the spread of infection over time when the population consists of individuals susceptible to infection, sick, immune to vaccination or past infection, and partially susceptible i.e. the population is assigned to four compartments of individuals. Actions taken and their impact on the dynamics of the population are important. It is precisely the analysis of the impact of the preventive diagnosis and treatment that is particularly interesting in the model. How the mathematical model covers such actions is presented in Section~\ref{section:sircmodel}. The model created in this way is then adapted for Monte Carlo analysis in~\ref{section:controlfunction} and the results obtained on this basis are found in~\ref{section:applicationcemethod}.
The analysis of computational experiments concludes this discussion in~\ref{chapter:results}.

\section{\label{chapter:sircwithcemethod} Optimization of control for \textbf{SIRC} model.
Let us focus the attention on the introduction of \textbf{SIRC} model and presentation of the logic behind using the \textbf{CE} method to determine functions that optimize the spread of epidemics. The ideology behind creation of the \textbf{SIRC} model and its interpretation will be presented in order to match the right parameters for calculating the cost functions.

	The \textbf{CE} method here is used to solve the variational deterministic problem. Solving such problems has long been undertaken with the help of numerical methods with a positive result. In the position \cite{ekeland1999convex} proposals of the route for variational problems for optimization problems in time are presented. The main focus was on the non-convex problems, which often occur with optimal control problems. In the book by Glowinski~\cite{glowinski2008lectures} a review of the methods for solving variational problems was made. And then \cite{mumford1989optimal} presented a complicated method of approximating functions for the problem optimization.

\subsection{\label{section:sircmodel}\textbf{SIRC} model.} The subject of the work is to solve the problem optimizing the spread of the disease, which can be modeled with the \textbf{SIRC} model. This model was proposed by \citeauthor{casagrandi2006sirc
	\cite{casagrandi2006sirc}. Its creation was intended to create a better model that would describe the course of influenza type A. Contrary to appearances, this is an important topic because the disease, although it is widely regarded as weak, is a huge problem for healthcare. In the US alone, the cost associated with the influenza epidemic in the season is estimated to exceed USD 10 billion (v.~\cite{klimov1999surveillance}) and the number of deaths is over 21,000 per season (v.~\cite{simonsen1997impact}).

	Various articles have previously proposed many different mathematical models to describe a pandemic of influenza type A. An overview of such models was made by Earn~\cite{earn2002ecology}. In general, it came down to combining SIR models with the use of cross-resistance parameters (cf. details in the papers by Andreasen et al.~\cite{AndLinLev1997:Dynamics} and Lin et al.~\cite{lin1999dynamics}). The authors of the article write that the main disadvantage of this approach is the difficulty of the analysis and calculations with a large number of strains. The results of their research showed that the classic SIR model cannot be used to model and study influenza epidemics. Instead, they proposed the extension of the SIR model to a new class C that will simulate a state between a susceptible state and a fully protected state. This extension aims to cover situations where vulnerable carriers are exposed to similar stresses as they had before. As a result, their immune system is stronger when fighting this disease.

The \textbf{SIRC} model divides the community into 4 groups:
	\begin{tableitems}\itemsep2pt \parskip3pt \parsep2pt
		\item S - persons susceptible to infection, who have not previously had contact with this disease and have no immune defense against this strain
		\item I - people infected with the current disease
		\item R - people who have had this disease and are completely immune to this strain
		\item C - people partially resistant to the current strain (e.g. vaccinated or those who have had a different strain)
	\end{tableitems}
	Figure \ref{fig:sirc1} contains a general scheme of this model. The four rhombus represent the four compartments of individuals, the movement between the compartments is indicated by the continuous arrows. The main advantage of this model over other SIR users is the fact that after recovering from a given strain, in addition to being completely resistant to this strain, they are also partly immune to a new virus that will appear later. This allows you to model the resistance and response of people in the group to different types of disease.
	\begin{figure}[H]
		\centering
		\includegraphics[width=0.6\linewidth]{SIRC_2.PNG}
		\caption{\label{fig:sirc1}Schematic representation of the \textbf{SIRC} model.}
	\end{figure}
     Parameters	$\alpha, \delta$ i $\gamma$ can be interpreted as the reciprocal of the average time spent by a person in order of ranges $I, R, C$. Parameters $\mu$ with indices $S,I,R,C $ represent the natural mortality in each group, respectively. In some versions of the model, additional mortality rates are considered for the group of infected people. Here we assume that this factor is not affected by the disease and we will denote it as $\mu$ in later formulas ($\mu_{S} = \mu_{I} = \mu_{R} =\mu_{C} = \mu$). The next one $\sigma$ is the likelihood of reinfection of a person who has cross-resistance while the parameter $\beta$ describes the contact indicator.

	The \textbf{SIRC} model is represented as a set of four ordinary differential equations. Let $S(t)$, $I(t)$ be the number of people in adequate compartments. We have (v. \cite{casagrandi2006sirc})
	\begin{equation} \label{eq:ode_1}
		\begin{aligned}
		\frac{dS}{dt} &= \mu(1-S(t))-\beta S(t)I(t)+ \gamma C(t), \\
		\frac{dI}{dt} &= \beta S(t)I(t) + \sigma \beta C(t)I(t)-(\mu+\alpha)I(t), \\
		\frac{dR}{dt} &= (1- \sigma) \beta C(t)I(t) + \alpha I(t) - (\mu+\delta)R(t),\\
		\frac{dC}{dt} &= \delta R(t) - \beta C(t)I(t) - (\mu+\gamma)C(t),
		\end{aligned}
	\end{equation}
	with initial conditions
	\begin{equation} \label{eq:ode_2}
		\begin{split}
		S(0)=S_{0}, \, I(0)=I_{0}, \, R(0)=R_{0}, \, C(0)=C_{0}
		\end{split}
	\end{equation}

	\subsection{\label{section:controlfunction}Derivation of optimization functions}
	\begin{comment}
	Features optimizing this model have been proposed  by  Iacoviello et al.~\cite{iacoviello2013optimal}. The author of the article suggesting the calculations presented in \cite{zaman2008stability}. Two functions have been suggested there. One that affects vulnerable people and the other who deals with sick people. To determine the optimal solution, a strategy described by Kamien et al.~\cite{kamien2012dynamic} was used. To apply them, you need to update the equations accordingly \eqref{eq:ode_1}. They look like this:
	\end{comment}
An approach to optimize this model was proposed by~\citeauthor{iacoviello2013optimal}~\cite{iacoviello2013optimal}. In this article one can find the suggestion concerning performing the calculations as outlined by \citeauthor{zaman2008stability} in \cite{zaman2008stability}. Two functions were proposed in the approach as the parameter or controls. One relates to the \emph{susceptible people} and the other describes the number of \emph{sick people}. The method described by \citeauthor{kamien2012dynamic}~\cite{kamien2012dynamic} was used to determine the optimal solution. In order to apply it, the set of equations~\eqref{eq:ode_1} should be updated accordingly. After these modifications, it has the following form:
	\begin{equation} \label{eq:ode_3}
	\begin{aligned}
		\frac{dS}{dt} &= \mu(1-S)-\beta SI+ \gamma C - g(S(t),u(t)), \\
		\frac{dI}{dt} &= \beta S(t)I(t) + \sigma \beta C(t)I(t)-(\mu+\alpha)I(t) - h(I(t),v(t)), \\
		\frac{dR}{dt} &= (1- \sigma) \beta C(t)I(t) + \alpha I(t) - (\mu+\delta)R(t) + g(S(t),u(t)) + h(I(t),v(t)),\\
		\frac{dC}{dt} &= \delta R(t) - \beta C(t)I(t) - (\mu+\gamma)C(t),
		\end{aligned}
	\end{equation}
	where
	\begin{equation} \label{eq:ode_4}
		\begin{aligned}
	g(S(t),u(t))&=\rho_{1}S(t)u(t),\\
	h(I(t),v(t))&=\rho_{2}I(t)v(t).
		\end{aligned}
	\end{equation}
	$u(t)$ represent the percentage of those susceptible who have been taken care of thanks to using control on population and  $v(t)$ represent the percentage of infected people with the same description, respectively. $\rho_{1}$ and $\rho_{2}$ are weights that optimize the proportion of given control options. Both functions $g(S(t),u(t))$ $h(I(t),v(t))$ and in order they can be interpreted as actions performed on people susceptible to the disease and infected. In addition, two new conditions related to optimization functions are added to the initial conditions.
	\begin{equation} \label{eq:ode_5}
	\begin{split}
	u_{min} \leqslant u(t) \leqslant u_{max}, \, v_{min} \leqslant v(t) \leqslant v_{max}
	\end{split}
	\end{equation}
	They describe the limits on what part of the population care can be given to. The smallest values $u_{min}$ and $v_{min}$ is zero.
	The existence of the solution is shown by Iacoviello et al.~\cite{iacoviello2013optimal} and they are functions that minimize the following cost index:
	\begin{equation} \label{eq:ode_6}

	J(u,v) = \int_{t_{1}}^{t_{2}}  \varphi(S(t),I(t),u(t),v(t))  dt
	= \int_{t_{1}}^{t_{2}} \left[ \alpha_{1}S(t) + \alpha_{2}I(t) + \frac{1}{2}\tau_{1}u^{2}(t) + \frac{1}{2}\tau_{2}v^{2}(t) \right] dt

	\end{equation}
	$\alpha_{1},\alpha_{2}$ are used to maintain the balance in the susceptible and infected group. $\tau_{1},\tau_{2}$ is interpreted as weighting in the cost index. $t_ {1}, t_ {2}$ determine the time interval. The square next to the functions $u(t)$ and $v(t)$ indicates increasing intensity of functions \cite{zaman2008stability}. Thus the function $J(u,v)$ reflects the human value of susceptible and infected persons, taking into account the growing value of funds used over a specified period of time.

    The objective function can be changed as long as there is a minimum. The \textbf{CE} method does not impose any restrictions here. However, to compare the results with \cite{iacoviello2013optimal}, the function proposed by them is used also here. There, the objective function must be square in order for the quadratic programming techniques to be used. Usually quadratic function is good in mathematics, but not in practice. Changing the objective function and impact of it can be considered further.

	\subsection{\label{section:applicationcemethod}Optimization of the epistemological model by  \textbf{CE} method.}
	\citeauthor{sani2007optimal} \cite{sani2007optimal} present the way in which the Cross-Entropy method can be used to solve the optimization of the epistemological model consisting of ordinary differential equations. The main task was to minimize the objective function $J(u)$ depending on one optimization function $u(t)$ over a certain set $U$ consisting of continuous functions $u$. The minimum can be saved as:
	\begin{equation}\label{eq:ce_1}
	\gamma^{*} =J(u^{*})=\min _{u\in U}J(u)
	\end{equation}
	Parameterizing the minimum problem looks as follows:
	\begin{equation}\label{eq:ce_2}
	\min _{c\in C}J(u_{c})
	\end{equation}
	where $u_{c}$ is a function from $U$, which is parameterized by a certain control vector $c\in \mathbb{R}^{m}$. Collection $C$ is a set of vectors $c$. It should be noted that the selected set $C$ should be big enough to get the enough precise solution. Then in such a set there are such $c^{*}$-- which will be the optimal control vector and  $\gamma^{*}$--optimal value, respectively.

    One way to parameterize a problem is to divide the time interval $[t_{1},t_{2}]$ at small intervals and using these intervals together with control vector points $c$ to define a function $u_{c}$. This function can be created by interpolating between points. Such interpolation can be done using e.g. the finite element method, finite difference method, finite volume method or cubic B-Spline . \citeauthor{fang2002finite} in \cite{fang2002finite} and \citeauthor{caglar2006} in \cite{caglar2006} show that all of this methods can be used to the two-point boundary value problems.  Here the \textbf{FEM}\label{FEM} method was used (v. \cite{PapGio2018:SFEM}). The results for the other methods have not been checked. The choice of the approximation method is a difficult and interesting issue, but this is not what is consider here. With this function $u_{c}$, it takes form
	\begin{equation}\label{eq:ce_3}
	u_{c}(t) = \sum_{t=0}^{n} c_{i}k_{i}(t)
	\end{equation}
	where
	\begin{equation}\label{eq:ce_4}
	  k_{i}(t) =
	\begin{cases}
	\frac{t-t_{i-1}}{t_{i}-t_{i-1}} & \text{for $t \in [t_{i-1},t_{i}]$}\\
	\frac{t_{i+1}-t}{t_{i+1}-t_{i}} & \text{for $t \in [t_{i},t_{i+1}]$}\\
	0 & \text{otherwise.}
	\end{cases}
	\end{equation}

	Now when $u_{c}$ is represented as a parameterized function, the value can be counted$J(u_{c})$ by solving the system of ordinary differential equations \eqref{eq:ode_3} e.g. by the Runge-Kutta method. The idea of using the \textbf{CE} method in this problem is to use a multi-level algorithm with generated checkpoints $c_{i}$, here, from a normal distribution. The \textbf{CE} method does not impose the distribution. Usually it is chosen from the family of density which is expected. Here, it's expected that there will be one high peak at the peak of the epidemic. Hence the normal distribution. Then, update the distribution parameters using the \textbf{CE} procedure using the target indicator as a condition for selecting the best samples. The calculations are carried out until the  empirical variance of optimal control function is smaller than the given $\epsilon$. This means that the function values are close to the optimal expected value  The idea of applying the \textbf{CE} method to the \textbf{SIRC} model is similar. The only difficulty is the fact that when optimizing the \textbf{SIRC} model there are two functions instead of one as in the case of the algorithm described here.
	\noindent In the form of an algorithm, it looks as follows: \\[2ex]
	Algorithm ( \emph{Modification of the \citeauthor{sani2007optimal}'s algorthm (v. ~\cite{sani2007optimal}) to two optimal functions case}):
	\begin{enumerate}\itemsep3pt \parskip3pt \parsep2pt
		\item Initialize: Choose $\mu^{(u)}_{0} = \{\mu_{i0}, i=0,\dots,n\}$, $\sigma^{(u)}_{0}=\{ \sigma_{i0},i=0,\dots,n \}$, $\mu^{(v)}_{0} = \{\mu_{i0}, i=0,\dots,n\}$ and $\sigma^{(v)}_{0}=\{ \sigma_{i0},i=0,\dots,n \}$, set $k=1$

		\item Draw: Generate a random samples $C_{1},\dots,C_{N} \sim N(\mu^{(u)}_{k-1},{\sigma^{(u)}_{k-1}}^{2})$ and $D_{1},\dots,D_{N} \sim N(\mu^{(v)}_{k-1},{\sigma^{(v)}_{k-1}}^{2})$ with $C_{m}=\{ C_{mi}, i=0,\dots,n \}$ and $D_{l}=\{ D_{li}, i=0,\dots,n \}$.\\[2ex]

		\item\label{AlgEvalJ1} Evaluate: For each control vector $C_{m}$ and $D_{m}$ evaluate the objective function $J(u_{C_{m}},u_{D_{l}})$ by solving the ODE system

		\item\label{AlgEvalJ2} Find the $p$ best performing samples, based on the values $\{J(u_{C_{m},D_{l}})\}$. Let $I$ be the corresponding set of indices.

		\item Update: for all $i=0,\dots,n$ let
		\begin{equation*}
		\begin{aligned}
		\hat{\mu_{ki}}^{(u)} = \frac{1}{p} \sum_{m\in I}^{}C_{mi};\qquad
		(\hat{\sigma_{ki}}^{(u)})^{2} = \frac{1}{p} \sum_{m\in I}^{}(C_{mi}-\mu^{(u)}_{ki}), \\
		\hat{\mu_{ki}}^{(v)} = \frac{1}{p} \sum_{l\in I}^{}D_{li}; \qquad
		(\hat{\sigma_{ki}}^{(v)})^{2} = \frac{1}{p} \sum_{l\in I}^{}(D_{li}-\mu^{(v)}_{ki}).
		\end{aligned}
		\end{equation*}

		\item Smooth: For a fixed smoothing parameter $ 0 < \alpha \leqslant 1$ let
		\begin{equation*}
		\begin{aligned}
		\hat{\mu_{k}}^{(u)} &= \alpha\mu_{k}^{(u)}+(1-\alpha)\hat{\mu_{k-1}}^{(u)};\qquad
		\hat{\sigma_{k}}^{(u)} &= \alpha\sigma_{k}^{(u)}+(1-\alpha)\hat{\sigma_{k-1}}^{(u)} \\
		\hat{\mu_{k}}^{(v)} &= \alpha\mu_{k}^{(v)}+(1-\alpha)\hat{\mu_{k-1}}^{(v)}; \qquad
		\hat{\sigma_{k}}^{(v)} &= \alpha\sigma_{k}^{(v)}+(1-\alpha)\hat{\sigma_{k-1}}^{(v)}
		\end{aligned}
		\end{equation*}

		\item Repeat 2–6 until $max_{i}\sigma_{ki}^{(u)}< \epsilon$ and $max_{i}\sigma_{ki}^{(v)}< \epsilon$ with $\epsilon = 10^{-5} $ . Let $L$ be the final iteration number. Return $\mu_{L}$ as an estimate of the optimal control parameter $c^{*}$
	\end{enumerate}
    The value of the  criterion $ J(\cdot, \cdot) $ is calculated in steps \ref{AlgEvalJ1} and \ref{AlgEvalJ2}. The selection of the function $\varphi$ allows to fit the cost function to the modeled case. Here, the criterion given by \eqref{eq:ode_6} is adopted.

	\section{\label{chapter:results}Description of the numerical results.}
	This section will present the results obtained using the Cross-Entropy method. For the correct comparison the same parameters for the \textbf{SIRC} model were used. These values are:
	\begin{equation*}
	\begin{aligned}
	\mu&=\frac{1}{75}[\text{year}^{-1}], \, &\gamma&=\frac{1}{2}[\text{year}^{-1}], \, &\alpha&=\frac{365}{5}[\text{year}^{-1}], \\
	\delta&=1[\text{year}^{-1}], \,
	&\beta &\approx 146 [\text{year}^{-1}], \, &\sigma&\approx 0.078.
	\end{aligned}
	\end{equation*}
	For optimization functions $u(t)$ and $v(t)$ the following restrictions have been applied:
	\begin{equation*}
	0 \leqslant u(t) \leqslant 0.9, \, 0 \leqslant v(t) \leqslant 0.9
	\end{equation*}
	Restrictions were proposed by \citeauthor{lenhart2007optimal}~\cite{lenhart2007optimal}). Their value was explained by the fact that the whole group cannot be controlled. In addition, the weights used for functions have values $\rho_{1}=2$ and $\rho_{2}=2$ \\
	Now, all that's left is to propose variable values for the objective function:
	\begin{equation*}
	\alpha_{1}=10^{-3}, \, \alpha_{2}=0.997, \, \tau_{1}=10^{-3}, \, \tau_{2}=10^{-3}.
	\end{equation*}

	The parameter values are adjusted using historical data like parameter mortality rate $\mu$, which is counted as average lifetime of a host or parameters $\alpha, \delta, \gamma$ mean the inverted time of belonging to the group $I, R$ and $C$. A detailed description of determining parameters along with their limitations is presented in the article by \citeauthor{casagrandi2006sirc}~\cite{casagrandi2006sirc}.

    \subsection{A remark about adjusting the parameters of the control determination procedure.} Here these parameters are given and describes the influenza A epidemics well, because of long historical data sets. However, this may not be the case, and then these parameters are subject to adjustment in the control determination procedure. An overview of applications and accuracy of calibration methods, which can be used for it is presented in the article by \citeauthor{hazelbag2020calibration}~\cite{hazelbag2020calibration}, which provides an overview of the model calibration methods. All parameters in the model are constant and independent of time, so methods which try to optimizes a goodness-of-fit (\textbf{GOF}\label{GOF}) can be use to this example. \textbf{GOF}\label{GOF} is a measure that assesses consistency between model the output and goals. As a result, it gives the best combination of parameters. Examples of such methods are Grid search, Nelder-Mead method, the Iterative, descent-guided optimisation algorithm, Sampling from a tolerable range. After finding the appropriate parameters, other algorithms like the profile likelihood method or Fisher information can be used to calculate the confidence intervals for these coefficients. If the epidemic described by the model consists of transition probabilities that cannot be estimated from currently available data, calibrations can be performed to many
	end points. Then \textbf{GOF} is measured as the mean percentage deviation of the values obtained at the endpoints (v. \cite{taylor2010methods}).

	\subsection{Proposed optimization methods in the model analysis.
	The results will be presented for two moments in the model: at the beginning and in the middle of the epidemic. This is initialized with other initial parameters. For the beginning of the epidemic, they are as follows:
	\begin{equation*}
	S_{0} = 1-I_{0}, \, I_{0}=10^{-6}, \, R_{0}=0, \, C_{0}=0.
	\end{equation*}
	And for the widespread epidemic:
	\begin{equation*}
	S_{0} = 0.99, \, I_{0}=5*10^{-3}, \, R_{0}=3*10^{-3}, \, C_{0}=2*10^{-3}.
	\end{equation*}
	The values in the model have been normalized for the entire population, and they add up to 1. The results for both models without the use of controls are shown in \ref{fig:sircsolutionwithout}. They were obtained using the Runge-Kutty method. The cost index in this case was in order $0.00799$ and $0.00789$ for the model of the beginning and the development of an epidemic.

In Sections~\ref{section:version1} and \ref{section:version2} two ways of calculating optimal functions $u(t)$ and $v(t)$ for the \textbf{SIRC} model will be considered. In the first version, the functions will be calculated separately. In the next both $u(t)$ and $v(t)$ will be counted at the same time. All versions will be presented at two moments of the epidemic: when it began and when it has already spread. The results will be compared with those obtained in the article by Iacoviello et al.~\cite{iacoviello2013optimal}, where the problem was solved using the sequential quadratic programming method using a tool from Matlab. Unfortunately, the article does not specify the cost index value for solving the obtained sequential quadratic programming method. Therefore, in the paper, there were attempts to recreate the form of the function $u(t)$ and $v(t)$ and the cost index obtained for them ($0.003308$ for the first situation and $0.006489$ for the second one) and the model solution.

	\begin{figure}[tbh!]
		\centering
		\includegraphics[width=0.75\linewidth]{SIRC_solution_without.PNG}
		\caption{\label{fig:sircsolutionwithout}Solutions for \textbf{SIRC} models without optimal control (in order the epidemic started and developed)}
	\end{figure}

	As can be seen in Figure \ref{fig:ceversion0} in the first situation, the control functions helped reduce peak infection, which was previously without control (Figure \ref{fig:sircsolutionwithout}) occurred between 2 and 4 months. Previously, the largest peak was around $20\%$ occurred between 2 and 4 months. Previously, the largest peak was around $10\%$. In the case of the second situation, the peak also decreased, but not very clearly (it is also visible in the value of the cost index, which for the first situation is $0.003308$, and for the other one $0.006489$). It can be seen that timely control is also very important to reduce the harmful effects of an epidemic. For more conclusions on how to apply control properly in this model, see \cite{iacoviello2013optimal}.
	\begin{figure}[tbh!]
		\centering
		\includegraphics[width=.75\linewidth]{CE_version_0.jpg}
		\caption{Values of control functions and solutions obtained for them. Solved by the sequential quadratic programming method}
		\label{fig:ceversion0}
	\end{figure}

	\subsection{\label{section:version1}\textbf{CE} method version 1.}\unskip In the first method, the algorithm shown at the end of the section \ref{section:applicationcemethod} has been applied twice. Once for the calculation of the optimal function $u(t)$ with a fixed $v(t)$, and then $v(t)$ with a fixed $u(t)$. In the next step, the results were combined and the result for the \textbf{SIRC} model was calculated using these two functions. This approach is possible due to the lack of a combined restriction on $u(t)$ and $v(t)$. Features are not dependent on each other with resources.

	The cost index was $0.003086$ for the start of epidemic and $0.006443$ a developed epidemic. The figure~\ref{fig:ceversion1} show the graphs of functions $u(t)$ and $v(t)$  along with a graph of solutions for both moments during the epidemic.

	\begin{figure}[tbh!]
		\centering
		\includegraphics[width=.7\linewidth]{CE_version_1.jpg}
		\caption{Values of control functions and solutions obtained for them. Solved by the cross-entropy method - version 1}
		\label{fig:ceversion1}
	\end{figure}

	For functions $u(t)$ the results are similar, only the decrease here is more linear. However, the solution of the \textbf{SIRC} model gives the same results. Function $v(t)$ is much more interesting. The \textbf{CE} method showed that it should have very low values, even close to $0$ throughout the entire period in both cases. However, this does not affect the cost index (it is even smaller than in the previous case) and the \textbf{SIRC} model. More on this subject will be included in the summary.

	\subsection{\label{section:version2}\textbf{CE} method version 2.}\unskip 	In the second case, it was necessary to change slightly the algorithm from Section~ \ref{section:applicationcemethod} to be able to use it for two control functions. In point 1 of the algorithm a variable is generated $C_m\sim N(\mu_{k},\sigma_{k}^{2})$, which serves to designate $u_{C_{m}}$, where $\mu_{k},\sigma_{k}$ are requested by the user. Here 2 $\mu$ will need to be initialized $\mu_{k}^{(u)},\mu_{k}^{(v)}$ and 2 $\sigma$: $\sigma_{k}^{(u)},\sigma_{k}^{(v)}$. Then designate $C_{1}^{(u)}, \dots ,C_{N}^{(u)} \sim N(\mu_{k}^{(u)},\sigma_{k}^{(u)2})$ and similarly $C_{1}^{(v)}, \dots ,C_{N}^{(v)} \sim N(\mu_{k}^{(v)},\sigma_{k}^{(v)2})$. Now $J(u,v)$ can be count without problems. Further the algorithm goes the same way, and in point 6 all 4 parameters $\mu_{k}^{(u)},\mu_{k}^{(v)},\sigma_{k}^{(u)},\sigma_{k}^{(v)}$ are updated. The graphs obtained for these functions are below.
	\begin{figure}[tbh!]
		\centering
		\includegraphics[width=.7\linewidth]{CE_version_2.jpg
		\caption{Values of control functions and solutions obtained for them. Solved by the cross-entropy method - version 2}
		\label{fig:ceversion2}
	\end{figure}

	In this case, we have very similar results as in the previous section. This can be seen in both the charts and the cost index values ($0.003044$ and $0.006451$). The only problem here is the need for a large sample to bring the results closer.

\subsection{\label{section:results}Comparison of results.} 	This section gathers the results from each version and compares them with each other. At first, the behavior of each group of the \textbf{SIRC} model was looked at and compared with the solution of the uncontrolled model. This is shown in Figures \ref{fig:scomparision}, \ref{fig:icomparision}, \ref{fig:rcomparision}, \ref{fig:ccomparision}. The results are presented only for the first situation, where the epidemic begins to spread, because there you can see the results of using control much better.
	\begin{figure}[H]
		\centering
		\includegraphics[height=0.25\textheight,width=.95\linewidth]{S_comparision.png
		\caption{Comparison of results obtained for group S among various methods}
		\label{fig:scomparision}
	\end{figure}

	\begin{figure}[H]
		\centering
		\includegraphics[height=0.25\textheight,width=.85\linewidth]{I_comparision.png}
		\caption{Comparison of results obtained for group I among various methods}
		\label{fig:icomparision}
	\end{figure}

The most interesting is Figure~\ref{fig:icomparision} comparing infected people. You can see here very well how functions $u(t)$ and $v(t)$ help in the fight against the epidemic. Let two versions of the \textbf{CE} method overlap. The functions obtained by the sequential quadratic programming method give slightly better results. However, they have a worse cost index (the comparison is presented in Table \ref{tab:summary}). This is probably due to the high value of the function $v(t)$, which increases the cost index. The question remains why there is such a discrepancy between the obtained function values $v(t)$ with both methods. In this article \cite{iacoviello2013optimal} weight and impact of functions $u(t)$ and $v(t)$ were compared. The chart in the article compares the results obtained for the application of both functions, one at a time and none. It turned out that the main influence on the number of infected has control over susceptible persons, i.e. the function $u(t)$. Choosing the optimal values for $v(t)$ is not very important for the solution of the model. Therefore, such discrepancies are possible when choosing optimal values.

\begin{figure}[H]
		\centering
		\includegraphics[height=0.27\textheight,width=.85\linewidth]{R_comparision.png}
		\caption{Comparison of results obtained for group R among various methods}
		\label{fig:rcomparision}
	\end{figure}

	\begin{figure}[tbh!]
		\centering
		\includegraphics[height=0.27\textheight,width=.85\linewidth]{C_comparision.png}
		\caption{Comparison of results obtained for group C among various methods}
		\label{fig:ccomparision}
	\end{figure}

	The table~\ref{tab:summary} contains a comparison of the cost index values and the duration of the algorithms for each method. The first value is when the epidemic started, and the second is when it has already developed. The article does not specify the algorithm time of the method used.
	\begin{table}[tbh!]
		\centering
		\begin{tabular}{|l|l|l|}
			\hline
			 Method & Cost index & Time \\\hline
			 \multirow{2}{*}{Results without control functions} & \multicolumn{1}{l|}{0.00799} & \multirow{2}{*}{-} \\\cline{2-2}
			 & \multicolumn{1}{l|}{0.00789} & \\\hline
			\multirow{2}{*}{Sequential quadratic programming method} & \multicolumn{1}{l|}{0.003308} & \multirow{2}{*}{-} \\\cline{2-2}
			& \multicolumn{1}{l|}{0.006489} & \\\hline
			\multirow{2}{*}{Cross-entropy method version 1} & \multicolumn{1}{l|}{0.003086} & \multirow{2}{*}{30s} \\\cline{2-2}
			& \multicolumn{1}{l|}{0.006443} & \\\hline
			\multirow{2}{*}{Cross-entropy method version 2} & \multicolumn{1}{l|}{0.003044} & \multirow{2}{*}{4m54s} \\\cline{2-2}
			& \multicolumn{1}{l|}{0.006451} & \\\hline
		\end{tabular}
		\caption{\label{tab:summary}Summary of cost indexes and times for all situations}
	\end{table}
	It can be seen that for each version the cost index values came out very similar. The differences can be seen in time. This is because in the first version of the \textbf{CE} method a smaller sample was simulated because both functions were considered separately. Fortunately, the functions were not closely related and it was possible to consider them separately.

\section{\label{chapter:summary}Summary.}\unskip
The study examined the possibility of using the \textbf{CE} method to determine the optimal control in the SIRC model. Similar models are being developed to model the propagation of malicious software in computer networks (v. \citeauthor{TayGubZhu2017:OICESHM}~\cite{TayGubZhu2017:OICESHM}). However, the number of works with this approach is still relatively small, although its universality should encourage checking its effectiveness. There are two control functions in the model, which when properly selected optimize the cost function. The \textbf{CE} method was used in two versions: considering the two functions separately and together. The results were then compared with those obtained by \citeauthor{iacoviello2013optimal}~\cite{iacoviello2013optimal} using routing of Matlab\footnote{Matlab is a paid environment with a number of functions. Unlike other environments and programming languages, additional features are created by developers in a closed environment and can only be obtained through purchase.}. The Cross-Entropy method had little problems when considering two functions at the same time. Due to the large number of possibilities, it was necessary to simulate a large sample, which significantly extended the algorithm's time. The second way came out much better. However, it only worked because the functions were not dependent on each other. In the opposite situation there is also a possibility to get result via \textbf{CE} results however, it will be necessary to change the algorithm and the dependence of how the functions depend on each other and add this to the algorithm when initiating optimization functions in the code. The same situation applies to  changing the objective function and the density used for approximation. Changing the objective function gives the possibility to use other methods. If it stays quadratic and the constraints are linear, quadratic programming techniques like the sequential quadratic programming method can still be used. If the objective function and the constraints stays convex, we use general methods from convex optimization. The \textbf{CE} method is based on well-known and simply classical rules and is therefore quite problem-independent. There are no rigid formulas and therefore it requires consideration for each problem separately. So it is very possible to reconsider it in another way, which can be interesting for further work. The \textbf{CE} method may also facilitate the analysis of modifications of epidemiological models related to virus mutation or delayed expression (v. \citeauthor{GubTayZhu2018:OCHMV}~\cite{GubTayZhu2018:OCHMV}, \citeauthor{KocGraLip2020:COVID}~\cite{KocGraLip2020:COVID}).
\vspace{6pt}

\authorcontributions{KSz gives an idea of application \textbf{CE} to SIRC model; MKS realizes implementation and simulation.}

\funding{KSz are thankful to Faculty of Pure and Applied Mathematics for research funding.}

\conflictsofinterest{The authors declare no conflict of interest.}

\abbreviations{\label{abbrAll}The following abbreviations are used in this article:\\
\begin{tabular}{@{}lp{0.8\linewidth}}
\textbf{CE}  &  Cross Entropy Method (v. page \pageref{IS::important})\\%[1ex]
\textbf{FEM} & the finite element methods (v. page \pageref{FEM})\\
\textbf{GOF} & a goodness-of-fit (v. page \pageref{GOF})\\
\textbf{IS}  & the importance sampling (v. page \pageref{IS::important})\\
\textbf{SIR} & The \textbf{SIR} model is one of the simplest \emph{compartmental} models. The letters means the number of \textbf{S}usceptible-\textbf{I}nfected-\textbf{R}emoved  (and immune) or deceased individuals.  (v. \cite{KerMcK1991:MTE1}, \cite{HarLobMak2014:Exact}).\\
\textbf{SIRC}& The \textbf{SIR} model with  the additional group of partially resistant to the current strain people: \textbf{S}usceptible - \textbf{I}nfectious - \textbf{R}ecovered - \textbf{C}ross-immune  (v. page~\pageref{section:sircmodel}).
\end{tabular}
}

redpajama_set_name: RedPajamaArXiv