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Home » Opinion » Middle powerplay
Middle powerplay
A coalescing of middle power interests when it comes to trade would allow safety-net pacts to be put in place that would allow these states to support each other if and when subjected to economic coercion. Similarly
Statesman News Service | August 3, 2022 9:41 am
(Illustration: iStock)
Middle powers in the Asia-Pacific including India are recalibrating their diplomatic engagement within and outside the region to insulate themselves from Sino-American strategic competition. The effort to influence rulemaking in terms of security, trade, and international law to protect their respective national interests is clearly discernible. According to Asia expert Stephen R Nagy, Director of Policy Studies at the Yokosuka Council for Asia-Pacific Studies, the change in the power balance associated with China's re-emergence as Asia's largest economy has once again brought forth concerns about the fallout of US-China friction (now heightened due to ongoing tensions over Taiwan) and raised questions about American leadership in the region among many of the USA-aligned ~ to a greater or lesser degree ~ middle powers such as Australia, Japan, Canada, and India.
Asia-Pacific middle powers are keeping a wary eye on Beijing which despite months of domestic turmoil over its zero-Covid policy, a pandemic-hit economy, the crackdown on tech companies, repression in Xinjiang, sabre-rattling on Taiwan et al has not budged an inch from its policy of fomenting instability in the region. Or, from the Chinese point of view, creating their own version of stability in the Asia-Pacific. China's continued aggression in the maritime domain, its implementation of policies designed to undermine the emerging digital economy of the region, and its coercive economic behaviour has gathered steam in recent months leaving middle powers especially vulnerable.
As Nagy puts it: "Therefore, the Asia-Pacific's middle powers are aligning to adapt to these changing dynamics and transforming their diplomacy and cooperation into 'neo-middle-power diplomacy'." But it's not all smooth sailing for the middle powers given they don't agree on everything and the varying intensity of their bilateral relationships with America. In this scenario, most experts believe the way forward for these states would be to actively seek out alignment partners both inside and outside the region based on a convergence of interests.
The sticking point, however, is likely to be the issue of US involvement; some Asia-Pacific nations would prefer it while others feel it should not be a prerequisite for alignment. But it is increasingly clear that the middle powers are focusing on key areas based on the synergy of their respective comparative advantages. Ideally, says Nagy, these would stress capability-based contributions, such as intelligence gathering, rather than the capacity of the resources available for cooperation. Regularising and institutionalising humanitarian/disaster-relief activities, intensifying maritime cooperation in the South China Sea, Taiwan Strait, and Indian Ocean, and joint transits in the Asia-Pacific are key areas in this respect.
A coalescing of middle power interests when it comes to trade would allow safety-net pacts to be put in place that would allow these states to support each other if and when subjected to economic coercion. Similarly, agreement on strengthening the architecture on free flow of data would provide a fillip to the emerging digital economies of the Asia-Pacific region.
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Shopping opportunities abound in Cancun, with a vast variety of shops ranging from modern malls and designer stores, to open-air markets and the trays of insistent street vendors. It's a city built on tourism, so Cancun's shops invariably accept credit cards and the opening hours tend to run from about 10am to 7pm during the week, and all morning on weekends.
Silver jewellery, traditional wooden statues (Alebrije), pottery, and fleece or cotton blankets are all popular souvenirs from Cancun. Visitors should ensure that they don't buy anything containing black coral, which is an endangered and protected species and could land you in some trouble at the airport.
Within the glitzy Hotel Zone, the Plaza la Fiesta is a popular shopping venue, and the El Zocalo bazaar is a great option for locally-produced arts and crafts. The main shopping strip of Paseo Kukulkan is lined with shops and souvenir stalls selling all the usual tourist trinkets and a few unexpected treasures. The Hotel Zone offers great quality and variety but the shopping experience can be somewhat soulless, in the sense that the stores tend to be international and there is little authentic local flavour on offer.
Mainland Cancun, commonly just called 'downtown', is less glamorous but much more affordable. Yaxchilan Avenue is a good place to start, and the market on Tulum Avenue is wonderful for those seeking out local crafts, particularly silverware. Travellers will be expected to bargain in the local markets.


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Q: Questions concerning PCI-E 2.1 vs. 2.0 My older PC's video card just kicked the bucket and has been BSODing. It has a PCI-E x16 slot. Newegg has video cards that are PCI-E x16, 2.0, 2.1, etc.
The slot on my ASUS P5N-E SLI motherboard is 16x. Can I use 16x 2.0 and 2.1 video cards in it? I'm not too familiar with how that works and couldn't find a solid resource on it through Google.

A: You can use a 2.1 card in a 2.0 slot or 1.0 slot it will work fine, I have a 2.1 card running in my desktop which has 2.0 right now. However if it is a 2.0 card in a 1.0 slot it will not be as fast, which is to be expected of course.

redpajama_set_name: RedPajamaStackExchange


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redpajama_set_name: RedPajamaGithub


The Blue Room Bar features turquoise booths, an old school jukebox, an eclectic mix of people from every end of the world and great prices.
I love this place . Just the look of it is so very cool - like the 50s or 60s in the best blue way.
There is a smoking section in the garden.


Very resistant and machine washable !
My mom always loved dogs. She wanted to be active in sports but didn't want to leave me alone . So she decided to sign up for some courses in agility and "Flyball" with me.
In doing so, she quickly realized that it was quite difficult to obtain interesting and stimulating toys for me.
Knowing that i loved the feel of fur and loved to play tug of war, she decided to combine both, and created a toy in faux fur for me. It became my favorite toy, and i would do anything for this reward.
But her biggest surprise was to see that all my friends would steal the tug away from me. With the growing demand the Schum-Tug Company was created in my name. Soon all my friends had their "SCHUMMI".
Obviously, we had found a winning formula for both doggie and his master.
They play together and they have fun !
The best toy, for your best friend !


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Vodní plyn je látka získávaná zplyňováním koksu, případně uhlí. Slouží buď jako palivo (pro svícení a vytápění v domácnostech i v průmyslu), nebo jako meziprodukt chemické výroby. Připravuje se tak, že se vodní pára vede přes koks rozžhavený na vysokou teplotu. Vodní plyn vzniká endotermickou reakcí
C + H2O → CO + H2

Složení
 Oxid uhelnatý - jedovatý
 Vodík - explozivní

Lowův proces
V roce 1873 si patentoval profesor Thaddeus S. C. Lowe postup výroby vodíku, který využíval dodatečnou konverzi vzniklého vodního plynu:
CO + H2O → CO2 + H2
Proces byl objeven při zkoumání procesů probíhajících při průchodu vysokotlaké páry vrstvou horkého uhlí, což je hlavní zdroj koksárenského plynu. Lowe tento proces zdokonalil tak, že uhlí zůstávalo v přehřátém stavu, což umožnilo velkokapacitní výrobu tohoto plynu. Proces poskytuje směs oxidu uhličitého a čistého vodíku, která je po ochlazení prováděna přes vodní páru. Poté získáme čistý vodík.

Reference

Anorganické směsi
Paliva
Voda
Vodíková technika

redpajama_set_name: RedPajamaWikipedia


The disease of obesity is now the most common disease experienced by humans. Improved well-being of life did not rule out the possibility of disease enters the human body, but instead cause other diseases arise. One such disease is obesity. Obesity is a condition where the body is the accumulation of excessive fat so overweight.
Mild obesity that those who are overweight 20-40%.
Obesity was that those with 41-100% overweight.
Severe obesity that those who are overweight by more than 100% and there is about 5% found in the fat people in the world.
However, we also need to notice that there is accumulation of fat that is still being reasonable. In women prone to accumulation of fat in the hips and buttocks so that his body into shape like a pear. In men tend to accumulation of fat in the abdomen so that her body shape like apples. But at certain times such as menopause, women may experience a body shape like apples and pear men. Well, they are at risk of obesity is to accumulate fat in the abdomen. Therefore, the picture of a pear body shape is actually better than the picture of the apple body shape.
Genetics is one of the causes of obesity. Genetic factors contribute 33% of the weight. But did you know that in fact not only the genes are passed on to offspring but also the lifestyle that has been accustomed to a family. So if parents tend to eat a lot of fat and the children were going to follow, not to mention because parents are also getting used like that.
2. Factors mind / psyche.
It can also affect a person's eating habits. For instance, could have been a reaction to emotional eating habits many acted on.
Environment includes lifestyle patterns or acted by someone. Well, these environmental factors that activity and food intake can be one of the most dangerous trigger obesity. Genetic factors can not be changed but can be adapted to environmental factors.
Obesity should not be underestimated! This disease can lead to other dangerous diseases such as diabetes, high blood pressure and stroke, heart disease and many other diseases. Therefore, a person who already have signs of obesity should immediately improve the pattern of life that is not affected by the other.
Before the obese would be better to watch. You can control yourself in the consumption of fatty foods especially those excessively. By doing enough physical activity so we can avoid the disease of obesity. Sometimes it takes a healthy diet even if it is nearly or already obese.


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Genomics saves West Vancouver woman's life after facing aggressive ovarian cancer
By Pauline Jeanneau on Oct 24, 2019
Pauline Jeanneau
Social Media Specialist , BC Cancer Foundation
See all blog posts by Pauline Jeanneau
Ovarian Cancer, Personalized Medicine Project, Personalized Onco-Genomics (POG)
Every three and a half hours, another woman is diagnosed with ovarian cancer in Canada. In B.C, more than 300 women will be diagnosed with a form of ovarian cancer this year.
In the winter of 2014, this statistic included West Vancouver's Candy Woodworth.
At her weekly Pilates class, Candy noticed she felt uncomfortable when doing a routine exercise. She went back to the class a week later, and noticed the same pain in her stomach and immediately made an appointment with her family doctor, who sent her for an urgent ultrasound.
"I hardly had my coat off when the phone rang and my doctor told me I had to come back immediately," Candy says. "My husband and I both went back, and it was then we found out I had ovarian cancer."
In March of 2015, Candy's treatment plan began with surgery, where her tumour was removed, followed by 18 rounds of chemotherapy.
A year later, at one of Candy's routine check-ups, her team discovered her cancer had returned, this time in her colon.
"My feelings at that point in time were still very positive – just get the job done, and that's exactly how my team at BC Cancer approached it," says Candy. She went in for surgery a second time.
Six months later, Candy's cancer returned again, in the exact same place in her colon. "It was very unusual for it to return to the exact same place," says Candy.
She underwent her third surgery.
With her cancer at an aggressive state, Candy was put forward as a prime candidate for an innovative, personalized cancer program at BC Cancer. She was put in touch with Dr. Anna Tinker, Provincial Gyne Oncology Tumour Group Chair and medical oncologist at BC Cancer.
The personalized onco-genomics (POG) program is BC Cancer's flagship study in precision medicine and the first program of its kind to use whole genome analysis to inform individual treatment planning for patients.
Dr. Anna Tinker
Through this program, Dr. Tinker and team found that Candy's tumour had a mutant BRCA gene, even though Candy herself did not carry the gene, and was able to match the tumour mutation to a clinical trial drug.
It's been two and a half years and Candy has been cancer-free since she began her journey on the POG program and started her tailored medication.
"The POG program and the fact they found a drug for me has given me my life back," says Candy. "It means everything – I now get the opportunity to be with my family and watch my grandchildren grow."
Learn more about the life-saving Personalized Onco-genomics program at: bccancerfoundation.com/POG
Innovative therapies are changing radiation treatment
The 2019 Inspiration Gala Raises $3 Million to Advance Genomics
Recent Guest Bloggers
Pauline Jeanneau (6)
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Announcing new guest blogger (52)
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Get the latest blog posts delivered to your inbox.

redpajama_set_name: RedPajamaCommonCrawl


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Govt. Higher Secondary Institute Botingoo is an educational institution located in Botingoo Sopore in the Baramulla District of Jammu and Kashmir, India and is one of the institutes in Jammu and Kashmir providing secondary education to the students of ambient areas as Watlab, Magraypora, Hathlangoo, Janwara, Mundji, Botingoo, Rampora and Rajpora. The institute is located in the village of Botingoo and has various infrastructural facilities. The institute has well equipped computer lab with Internet facility.  The school publishes an annual school magazine Al-Noor in which all the students as well as teachers share their own views. It also provides opportunities to its students to participate in various inter-district and inter-state tournaments. The students of the institute have won trophies and medals in different tournaments.

See also
Government Degree College Sopore

References

Baramulla district
High schools and secondary schools in Jammu and Kashmir
Sopore


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Home News U.S. Administers 362.7 Million Doses of COVID-19 Vaccines- CDC
U.S. Administers 362.7 Million Doses of COVID-19 Vaccines- CDC
The United States has administered 362,657,771 doses of COVID-19 vaccines in the country as of Sunday morning and distributed 428,531,345 doses, the U.S. Centers for Disease Control and Prevention said on Sunday.
Those figures are up from the 361,684,564 vaccine doses the CDC said had gone into arms by Saturday out of 428,506,065 doses delivered.
The agency said 201,425,785 people had received at least one dose while 170,821,621 people are fully vaccinated as of Sunday.
The CDC tally includes two-dose vaccines from Moderna (MRNA.O) and Pfizer/BioNTech (PFE.N), as well as Johnson & Johnson's (JNJ.N) one-shot vaccine as of 6:00 a.m. ET on Sunday.


text:
Vessel Name : GD
Operator : GIRARDEAU STEVEDORES
Vessel Type : Other
City : SCOTT CITY
Height : 9.6 Feet
EQUIP1 : CRANE
GIRARDEAU STEVEDORES
Area of Operation : CHESTER, IL MISSISSIPPI UPPER MILE 167.5 TO CARUTHERSVILLE, MO LOWER MISSISSIPPI MILE 849
Principal Commodity : CRANE
Graney Name President General Dynamics Electric Boat
General Dynamics (NYSE: GD) appointed General Dynamics NASSCO president Kevin Graney as president of General Dynamics Electric Boat, effective October 1. David J. Carver, NASSCO's vice president and general manager of repair, will succeed Graney as president of General Dynamics NASSCO. Jeffrey S. Geiger, who has served as president of Electric Boat since 2013, will retire effective September 30.
GD NASSCO Wins $1.6 Billion Navy Contract
General Dynamics NASSCO, a subsidiary of General Dynamics (NYSE: GD), was awarded a contract by the U.S. Navy worth up to $1.6 billion for the construction of the sixth and seventh ships of the Expeditionary Sea Base (ESB) program, as well as an option for ESB 8.The contract, announced by the Department of Defense today, provides $1.
GD NASSCO to Christen Matson Containership
General Dynamics NASSCO will hold a special event to christen and launch the containership Lurline on Saturday, June 15. Lurline is an 870-foot-long, 3,500 TEU, 44,200 dwt, combination containership/RoRo vessel built for Matson, Inc. The ship is the largest containership ever built in San Diego.Lurline has liquefied natural gas-capable engines and is designed for energy efficiency.
Meet PoBo: Floating Platform for Disaster Relief
GD European Land Systems develops floating platform for civil protection and disaster reliefGeneral Dynamics European Land Systems-Bridge Systems won a contract to develop a multi-functional, modular floating platform known as Pontoon Boat (PoBo). The project was awarded under the Innovation Program for the Support of Diversification Strategies of Defense Companies in Civil Security Technologies
GD Bags $1.6Bln Shipbuilding Contract
General Dynamics NASSCO, a subsidiary of General Dynamics, has been awarded a contract by the U.S. Navy worth up to $1.6 billion for the construction of the sixth and seventh ships of the Expeditionary Sea Base (ESB) program, as well as an option for ESB 8.According to the Department of Defense, the contract provides $1.
Hepburn and Sons Secures NSRP Award
The National Shipbuilding Research Program (NSRP) has awarded Hepburn and Sons LLC with the R&D Insulated Bus Pipe (IBP) Standard Interface to Naval Electrical Equipment.The team consist of Hepburn and Sons LLC, TEFELEN, NSWCPD, GD-Bath Iron Works, RSL Fiber, AeroNav, Staubli, FSU CAPS, KATO Engineering, Northrop Grumman Sykesville, General Atomics, Rolls Royce, ABS, ABB, DRS, and Roxtec.


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Global stability of a diffusive and delayed virus dynamics model with Crowley-Martin incidence function and CTL immune response
Chengjun Kang1,2,
Hui Miao2,
Xing Chen1,
Jiabo Xu1 &
Da Huang1
In this paper, a diffusive and delayed virus dynamics model with Crowley-Martin incidence function and CTL immune response is investigated. By constructing the Lyapunov functionals, the threshold conditions on the global stability of the infection-free, immune-free and interior equilibria are established if the space is assumed to be homogeneous. We show that the infection-free equilibrium is globally asymptotically stable if the basic reproductive number \(R_{0}\leq1\); the immune-free equilibrium is globally asymptotically stable if the immune reproduction number and the basic reproduction number satisfy \(R_{1}\leq1< R_{0}\); the interior equilibrium is globally asymptotically stable if \(R_{1}>1\).
Nowadays, virus infection is related to the global health problems. Many diseases which are caused by viruses, such as hepatitis B virus (HBV), human immunodeficiency virus (HIV), hepatitis C virus (HCV), have drawn the attention of researchers. Based on the virus infection models proposed in [1–18], several different mathematical models which are valuable to obtaining comprehensive views to the virus dynamics have been investigated, for example, in a form of ordinary differential equations (ODEs) [12, 19–21], delayed differential equations (DDEs) [1–5, 7, 9, 11, 22], partial differential equations (PDEs) [8, 22–28] and fractional-order differential equations (FODEs) [15–18]. Nowak and Bangham [19] pointed out the basic virus infection model which plays a critical role in understanding the virus replication dynamics in vivo. They considered the following basic mathematical model for uninfected host cells u, infected host cells w, free virus v and the magnitude of the CTL response z:
$$\begin{aligned}& {\frac{du(t)}{dt}}=\lambda-du(t)-\beta u(t)v(t), \\& {\frac{dw(t)}{dt}}=\beta u(t)v(t)-h w(t)-pw(t)z(t), \\ & \\ & {\frac{dv(t)}{dt}}=kw(t)-\mu v(t), \\& {\frac{dz(t)}{dt}}=cw(t)z(t)-qz(t), \end{aligned}$$
where the uninfected host cells are produced at rate λ, die at rate d and become infected at rate β. Infected host cells die at rate h and are killed by the CTL response at rate p. Free virus is produced from infected cells at rate k and is removed at rate μ. The magnitude of the CTL response, which expands in response to viral antigen derived from infected cells at rate c, decays in the absence of antigenic stimulation at rate q.
It should be mentioned here that, in model (1), the rate of infection is assumed to be bilinear, that is, \(\beta uv\). However, this assumption is not biologically sensible all the time. Recently, many researchers have performed the virus dynamics models with Crowley-Martin infection rate (see [20, 21]). The Crowley-Martin type of functional response, that is, \(\frac{\beta uv}{(1+au)(1+bv)}\), was introduced by Crowley and Martin in [29], where a, b are constants. Particularly, when \(a=0\), \(b=0\), the Crowley-Martin infection rate becomes bilinear infection rate. Thus, it is necessary to study virus infection model with Crowley-Martin infection rate. In addition, based on the epidemiological background, time delays play a critical role in the virus infection model. To incorporate the intracellular phase of the virus life-cycle, we assume that virus production occurs after the virus entry by the intracellular delay \(\tau_{1}\). The recruitment of virus producing cells at time t is given by the number of uninfected cells that were newly infected at time \(t-\tau_{1}\) and are still alive at time t (see [2, 5, 9, 11]). The constant m is assumed to be the death rate for newly infected cells during time period \([t-\tau_{1},t]\). \(e^{-m \tau_{1}}\) denotes the surviving rate of infected cells during the delay period. Virus replication delay \(\tau_{2}\) represents the time necessary for the newly produced viruses to become mature and then infectious (see [10, 14, 22]). The constant n is assumed to be the death rate of a new virus during time period \([t-\tau_{2},t]\). \(e^{-n \tau_{2}}\) denotes the surviving rate of a virus during the delay period.
Until now, there has been a large number of works about virus infection models which considered delays, but in many biological systems, the species under consideration may disperse spatially as well as evolve in time (see [30]). As a matter of fact, many models ignored the spatial mobility of cells and viruses. Recently, many authors argued that the virus moves freely in body and follows the Fickian diffusion (see [31]) and investigated the global stability properties of virus infection models with diffusion in [8, 22–28, 31]. But the research is relatively small, a lot of work needs to be further done to provide theoretical evidence for controlling disease.
In this paper, motivated by the work of [6, 22, 23], we further neglect the mobility of susceptible cells, infected cells and immune cells and consider a delayed virus infection model with Crowley-Martin infection rate and spatial diffusion:
$$ \begin{aligned} &{\frac{\partial u}{\partial t}}=\lambda -du(x,t)-{ \frac{\beta u(x,t)v(x,t)}{ (1+au(x,t) ) (1+bv(x,t) )}}, \\ &{\frac{\partial w}{\partial t}}=e^{-m\tau_{1}}{\frac{\beta u(x,t- \tau_{1})v(x,t-\tau_{1})}{ (1+au(x,t- \tau_{1}) ) (1+bv(x,t-\tau_{1}) )}}-hw(x,t)-pw(x,t)z(x,t), \\ &{\frac{\partial v}{\partial t}}= D\Delta v(x,t)+ke ^{-n\tau_{2}}w(x,t- \tau_{2})-\mu v(x,t), \\ &{\frac{\partial z}{\partial t}}=cw(x,t)z(x,t)-qz(x,t) \end{aligned} $$
for \(t>0\), \(x\in\Omega\subset\mathbb{R}^{n}\) with the initial conditions
$$ \begin{aligned} &u(x,\theta)=\phi_{1}(x, \theta)\geq 0,\qquad w(x,\theta)=\phi_{2}(x, \theta)\geq0, \\ &v(x,\theta)=\phi_{3}(x,\theta)\geq0, \qquad z(x,\theta)= \phi_{4}(x, \theta)\geq0,\quad x\in\bar{\Omega}, \theta\in [-\tau ,0], \end{aligned} $$
and the homogeneous Neumann boundary conditions
$$ \frac{\partial v}{\partial\vec{n}}=0, \quad t>0, x\in\partial\Omega, $$
where \(u(x, t)\), \(w(x, t)\), \(v(x, t)\) and \(z(x, t)\) represent the densities of uninfected cells, infected cells, free virus and immune cells at location x and time t, respectively. The Laplacian operator and the diffusion coefficient are denoted by Δ and D, respectively. \(\tau=\max\{\tau_{1}, \tau_{2}\}\), Ω is a connected, bounded domain in \(\mathbb{R}^{n}\) with smooth boundary ∂Ω. \(\frac{\partial}{\partial\vec{n}}\) denotes the outward normal derivative on ∂Ω. \(\phi_{i}(x,\theta)\) (\(i=1,2,3,4\)) are nonnegative and Hölder continuous in \(\bar{\Omega}\times[-\tau,0]\). The boundary conditions in (4) imply that the virus particles do not move across the boundary ∂Ω.
In this paper, the purpose is to investigate the dynamical properties of model (2), expressly the stability of equilibria. Our paper is organized as follows. In the next section, we discuss the positivity and boundedness of solutions, the threshold values and the existence of equilibria of model (2). In Section 3, by constructing Lyapunov functionals, we establish global stability of all equilibria of model (2). In Section 4, we further illustrate the dynamical behavior by numerical simulations. In the last section, we give brief conclusions.
Positivity, boundedness and equilibrium
In this section, our main purpose is to prove the existence, positivity and boundedness of solutions of model (2).
For any given initial data satisfying condition (3), there exists a unique solution of model (2) defined on \([0,+\infty)\), and this solution remains nonnegative and bounded for all \(t\geq0\).
By standard existence theory [32–34], it is easy to establish the local existence of the unique solution \((u(x,t), w(x,t), v(x,t),z(x,t))\) of model (2) for \(x\in\overline{\Omega}\) and \(t\in[0,T_{\max}]\), where \(T_{\max}\) is the maximal existence time for solution of the model (2).
It is not hard to see that \(\textbf{0}=(0,0,0,0)\) and \(\textbf{M}=(M _{1}, M_{2}, M_{3}, M_{4})\) are a pair of coupled lower-upper solutions to model (2), where \(M_{1}\), \(M_{2}\), \(M_{3}\) and \(M_{4}\) satisfy
$$ \begin{aligned} &M_{1}=\max \biggl\{ {\frac{\lambda}{d}}, \sup_{-\tau\leq s\leq0} \bigl\Vert \phi_{1}(\cdot, s) \bigr\Vert _{C(\overline{\Omega},R)} \biggr\} , \\ &M_{2}=\max \biggl\{ {\frac{\beta M_{1}}{lb}}e^{-m\tau_{1}}, \sup _{-\tau\leq s\leq0} \bigl\Vert \phi_{1}(\cdot, s) \bigr\Vert _{C(\overline{\Omega},R)} \biggr\} , \\ &M_{3}=\max \biggl\{ {\frac{k\beta M_{1}}{\mu lb}}e^{-(m\tau_{1}+n\tau _{2})}, \sup _{-\tau\leq s\leq0} \bigl\Vert \phi_{1}( \cdot, s) \bigr\Vert _{C(\overline{\Omega},R)} \biggr\} , \\ &M_{4}=\max \biggl\{ {\frac{c\beta M_{1}}{plb}}e^{-m\tau_{1}}, \sup _{-\tau\leq s \leq0} \bigl\Vert \phi_{1}(\cdot, s) \bigr\Vert _{C(\overline{\Omega},R)} \biggr\} , \end{aligned} $$
and \(l=\min\{q,h\}\). From the comparison principle [35], we have \(0\leq u(x,t)\leq M_{1}\), \(0\leq w(x,t)\leq M_{2}\), \(0\leq v(x,t) \leq M_{3}\), \(0\leq z(x,t)\leq M_{4}\) for \(x\in\overline{\Omega}\) and \(t\in[0, T_{\max}]\). Then \(u(x,t)\), \(w(x,t)\), \(v(x,t)\) and \(z(x,t)\) are bounded on \(\overline{\Omega}\times[0, T_{\max})\). By the standard theory for semilinear parabolic systems [36, 37], we can deduce that \(T_{\max}=+\infty\). This completes the proof.
In the following, we will discuss the existence of equilibria of model (2). Model (2) always has an infection-free equilibrium \(E_{0}=(u_{0}, 0, 0, 0)\), where \(u_{0}={\frac{\lambda}{d}}\). The basic reproduction number is given by
$$ R_{0}=\frac{\beta k \lambda}{\mu h(d+a\lambda) e^{m\tau_{1}+n\tau _{2}}}. $$
We can rewrite \(R_{0}\) in the following form:
$$ R_{0}=k\cdot\frac{1}{\mu}\cdot e^{-n\tau_{2}}\cdot \frac{\beta \cdot\frac{\lambda}{d}}{1+a\cdot\frac{\lambda}{d}}\cdot e^{-m\tau _{1}}\cdot\frac{1}{h}. $$
Here, k is the rate of new virus particles produced by infected cells, \(\frac{1}{\mu}\) is the surviving period of virus, \(e^{-n\tau_{2}}\) is the surviving rate of a new virus in time period \([t-\tau_{2}, t]\), \(\frac{\beta\cdot\frac{\lambda}{d}}{1+a\cdot\frac{\lambda}{d}}\) denotes the newly infected cells which are infected by the first virus, \(e^{-m\tau_{1}}\) is the surviving rate of newly infected cells in time period \([t-\tau_{1}, t]\), and \(\frac{1}{h}\) is the surviving period of infected cells. Therefore, we easily see that \(R_{0}\) denotes the average number of the free viruses released by the infected cells which are infected by the first virus.
If \(R_{0}>1\), there exists a unique immune-free equilibrium \(E_{1}=(u_{1},w_{1},v_{1},0)\), where \(u_{1}\) is a positive root of the following equation:
$$ abkdu^{2}+ \bigl(\beta k+bkd-abk\lambda-a\mu h e^{m\tau_{1}+n\tau_{2}} \bigr)u- \bigl(bk \lambda+\mu h e^{m\tau_{1}+n\tau_{2}} \bigr)=0, $$
and \(w_{1}={\frac{\mu e^{n\tau_{2}}}{kb}}(R^{*}-1)\), \(v_{1}={\frac{1}{b}}(R ^{*}-1)\) if and only if
$$ R^{*}={\frac{\beta ku_{1}}{h\mu(1+au_{1})e^{m\tau_{1}+n\tau_{2}}}}>1. $$
Also, an immune response reproduction number is
$$ R_{1}={\frac{c}{q}}w_{1}={\frac{c\mu e^{n\tau_{2}}}{qkb}} \bigl(R^{*}-1 \bigr). $$
Note that when \(R_{0}>1\) model (2) has a unique immune-free equilibrium \(E_{1}=(u_{1},w_{1},v_{1},0)\). This shows that virus infection is successful and the numbers of free viruses and infected cells at equilibrium \(E_{1}\) are \(v_{1}\) and \(w_{1}\), respectively. Furthermore, we have that \(\frac{1}{q}\) is the average life-span of CTL cells and c is the rate at which the CTL response is produced. Hence, \(R_{1}\) denotes the average number of the CTL immune cells activated by infected cells when virus infection is successful.
If \(R_{1}>1\), then model (2) has a unique interior equilibrium \(E_{2}=(u_{2}, w_{2}, v_{2}, z_{2})\), where \(u_{2}\) is a positive root of the following equation:
$$\begin{aligned}& \bigl(a\mu cd+abkqde^{-n\tau_{2}} \bigr)u^{2}+ \bigl(\beta kqe^{-n\tau_{2}}+d\mu c+bkqde ^{-n\tau_{2}}-a\mu c\lambda -abkq\lambda e^{-n\tau_{2}} \bigr)u \\& \quad{}- \bigl(\lambda\mu c+bkq\lambda e^{-n\tau_{2}} \bigr)=0, \end{aligned}$$
$$ w_{2}={\frac{q}{c}}, \qquad v_{2}={ \frac{ke^{-n\tau_{2}}}{\mu}}w_{2}, \qquad z_{2}={\frac{\lambda-du_{2}-hw_{2}e^{m\tau_{1}}}{pw_{2}e^{m\tau_{1}}}}. $$
Stability analysis
In this section, we investigate the global stability of equilibria for model (2), namely, infection-free equilibrium \(E_{0}\), immune-free equilibrium \(E_{1}\) and interior equilibrium \(E_{2}\) of model (2), respectively. For the sake of convenience, we let \(y=y(x,t)\) and \(y_{\tau_{i}}=y(x,t-\tau_{i})\) for any \(y\in\{u, w, v, z\}\) and \(i\in{1,2}\).
Stability of equilibrium \(E_{0}\)
If \(R_{0} \leq1\), then the infection-free equilibrium \(E_{0}\) of model (2) is globally asymptotically stable.
Construct a Lyapunov functional
$$ W(t)= \int_{\Omega} \bigl(W_{1}(x,t)+W_{2}(x,t) \bigr)\,\mathrm{d}x, $$
$$\begin{aligned}& W_{1}(x,t)={\frac{u_{0}}{1+au_{0}}}\biggl({\frac{u}{u_{0}}}-1-\ln{ \frac {u}{u_{0}}}\biggr)+e^{m\tau_{1}}w+ {\frac{he^{m\tau_{1}+n\tau_{2}}}{k}}v+{ \frac{pe^{m\tau_{1}}}{c}}z, \\& W_{2}(x,t)= \int_{0}^{\tau_{1}} {\frac{\beta u_{\theta}v_{\theta }}{(1+au_{\theta}) (1+bv_{\theta})}} \,\mathrm{d} \theta+ he^{m\tau_{2}} \int_{0}^{\tau_{2}} w_{\theta}\, \mathrm{d}\theta. \end{aligned}$$
$$\begin{aligned}& {\frac{\partial W_{1}(x,t)}{\partial t}}+{\frac{\partial W_{2}(x,t)}{\partial t}} \\& \quad= {\frac{1}{1+au_{0}}}\biggl(1-{\frac{u_{0}}{u}}\biggr){ \frac{\partial u}{\partial t}}+e^{m\tau _{1}}{\frac{\partial w}{\partial t}}+{\frac{he^{m\tau_{1}+n\tau _{2}}}{k}} { \frac{\partial v}{\partial t}}+{\frac{pe^{m\tau _{1}}}{c}} {\frac{\partial z}{\partial t}} \\& \quad\quad{}+{\frac{\beta uv}{(1+au) (1+bv)}}-{\frac{\beta u_{\tau _{1}}v_{\tau_{1}}}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} +he^{m\tau _{2}}w-he^{m\tau_{2}}w_{\tau_{2}} \\& \quad= {\frac{1}{1+au_{0}}}\biggl(1-{\frac{u_{0}}{u}}\biggr) \biggl(\lambda -du-{\frac{\beta uv}{(1+au) (1+bv)}} \biggr)+{\frac{he^{m\tau_{1}+n\tau _{2}}}{k}}D \Delta v \\& \quad\quad{}-{\frac{h\mu e^{m\tau_{1}+n\tau_{2}}}{k}}v-{\frac {pqe^{m\tau_{1}}}{c}}z+{\frac{\beta uv}{(1+au) (1+bv)}} \\& \quad= -{\frac{d(u-u_{0})^{2}}{u(1+au_{0})}}- {\frac{1}{1+au_{0}}} {\frac{\beta uv}{(1+au) (1+bv)}}+{ \frac {1}{1+au_{0}}} {\frac{\beta u_{0}v}{(1+au) (1+bv)}} \\& \quad\quad{}+{\frac{\beta uv}{(1+au) (1+bv)}}-{\frac{h\mu e^{m\tau _{1}+n\tau_{2}}}{k}}v- {\frac{pq}{c}}e ^{m\tau_{1}}z+{\frac{he^{m\tau_{1}+n\tau _{2}}}{k}}D\Delta v \\& \quad= -{\frac{d(u-u_{0})^{2}}{u(1+au_{0})}}+ {\frac{1+au}{1+au_{0}}} {\frac{\beta u_{0}v}{(1+au) (1+bv)}}-{ \frac{h \mu e^{m\tau_{1}+n\tau_{2}}}{k}}v \\& \quad\quad{} -{\frac{pq}{c}}e^{m\tau_{1}}z+{\frac{he^{m\tau _{1}+n\tau_{2}}}{k}}D\Delta v. \end{aligned}$$
Calculating the derivative of \(W(t)\) along the positive solution of model (2), we get
$$\begin{aligned} {\frac{dW(t)}{dt}}&=- \int_{\Omega} {\frac{d(u-u_{0})^{2}}{u(1+au_{0})}}\,\mathrm{d}x+ \int_{\Omega}{\frac{\beta u_{0}v}{(1+au_{0}) (1+bv)}}\,\mathrm{d}x \\ &\quad{}- \int_{\Omega}{\frac{h\mu e^{m\tau_{1}+n\tau_{2}}}{k}}v\,\mathrm{d}x- \int_{\Omega} {\frac{pqe^{m\tau_{1}}}{c}}z\,\mathrm{d}x \\ &\quad{}+ \int_{\Omega} {\frac{he^{m\tau_{1}+n\tau_{2}}}{k}}D\Delta v\,\mathrm{d}x \\ &=- \int_{\Omega} {\frac{d(u-u_{0})^{2}}{u(1+au_{0})}}\,\mathrm{d}x + \int_{ \Omega} {\frac{h\mu e^{m\tau_{1}+n\tau_{2}}v}{k(1+bv)}}(R_{0}-1) \, \mathrm{d}x \\ &\quad{}- \int_{\Omega} {\frac{h\mu be^{m\tau_{1}+n\tau_{2}}}{k(1+bv)}}v^{2} \, \mathrm{d}x - \int_{\Omega} {\frac{pqe^{m\tau_{1}}}{c}}z\,\mathrm{d}x \\ &\quad{}+ \int_{\Omega} {\frac{he^{m\tau_{1}+n\tau_{2}}}{k}}D\Delta v\,\mathrm{d}x. \end{aligned}$$
Owing to the divergence theorem and the homogeneous Neumann boundary conditions (4), we obtain
$$ \int_{\Omega} \Delta v\,\mathrm{d}x= \int_{\partial\Omega} \frac{ \partial v}{\partial\vec{n}}\,\mathrm{d}x=0. $$
Thus, we further have
$$\begin{aligned} {\frac{dW(t)}{dt}}&=- \int_{\Omega} {\frac{d(u-u_{0})^{2}}{u(1+au_{0})}}\,\mathrm{d}x + \int_{ \Omega} {\frac{h\mu e^{m\tau_{1}+n\tau_{2}}v}{k(1+bv)}}(R_{0}-1) \, \mathrm{d}x \\ &\quad{}- \int_{\Omega} {\frac{h\mu be^{m\tau_{1}+n\tau_{2}}}{k(1+bv)}}v^{2} \, \mathrm{d}x - \int_{\Omega} {\frac{pqe^{m\tau_{1}}}{c}}z\,\mathrm{d}x. \end{aligned}$$
Therefore, \(\frac{dW(t)}{dt}\leq0\) if \(R_{0} \leq1\). We have \(\frac{dW(t)}{dt}= 0\) if and only if \(u=u_{0}\), \(w=0\), \(v=0\) and \(z=0\). It follows that the largest invariant set \(\{(u,w,v,z)\in R ^{4}_{+}: \frac{dW(t)}{dt}=0\}\) is the singleton \(E_{0}\). By using LaSalle's invariance principle [30], we see that the equilibrium \(E_{0}\) of model (2) is globally asymptotically stable when \(R_{0}\leq1\). □
If \(R_{1}\leq1< R_{0}\), then the immune-free equilibrium \(E_{1}\) of model (2) is globally asymptotically stable.
Let \(G(x,t)=g(x,t)-1-\ln g(x,t)\). We have that \(G(x,t)\geq0\) for all \(g(x,t)>0\) and \(G(x,t)=0\) if and only if \(g(x,t)=1\). Define a Lyapunov functional
$$ V(t)= \int_{\Omega} \bigl(V_{1}(x,t)+hw_{1}V_{2}(x,t) \bigr)\,\mathrm{d}x, $$
$$\begin{aligned}& V_{1}(x,t)= e^{-m\tau_{1}} \biggl(u-u_{1}- \int_{u_{1}}^{u}{\frac{1+a\theta}{1+au_{1}}} { \frac{u_{1}}{\theta}} \,\mathrm{d}\theta \biggr) +w_{1}G\biggl( { \frac{w}{w_{1}}}\biggr) +{\frac{he^{n\tau_{2}}}{k}}v_{1}G\biggl({ \frac{v}{v_{1}}}\biggr) +{\frac{p}{c}}z, \\& V_{2}(x,t)= \int_{0}^{\tau_{1}}G \biggl({\frac{e^{-m\tau_{1}}\beta u_{\theta }v_{\theta}}{hw_{1}(1+au_{\theta}) (1+bv_{\theta})}} \biggr) \,\mathrm {d}\theta+ \int_{0}^{\tau_{1}}G\biggl({\frac{w_{\theta}}{w_{1}}}\biggr)\, \mathrm{d}\theta. \end{aligned}$$
$$\begin{aligned}& {\frac{\partial V_{1}(x,t)}{\partial t}} = e^{-m\tau_{1}}\biggl(1-{\frac {1+au}{1+au_{1}}} { \frac{u_{1}}{u}}\biggr){\frac{\partial u}{\partial t}} +\biggl(1- {\frac{w_{1}}{w}} \biggr){\frac{\partial w}{\partial t}} \\& \hphantom{{\frac{\partial V_{1}(x,t)}{\partial t}} =}+{\frac{he^{n\tau_{2}}}{k}}\biggl(1-{\frac{v_{1}}{v}}\biggr){\frac{\partial v}{\partial t}} +{ \frac{p}{c}} {\frac{\partial z}{\partial t}}, \\& {\frac{\partial V_{2}(x,t)}{\partial t}}= {\frac{e^{-m\tau_{1}}\beta uv}{hw_{1}(1+au) (1+bv)}}-{\frac{e^{-m\tau_{1}}\beta u_{\tau_{1}}v_{\tau _{1}}}{hw_{1}(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \\& \hphantom{{\frac{\partial V_{2}(x,t)}{\partial t}}=}+\ln{\frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau_{1}}) (1+bv_{\tau _{1}})}} {\frac{(1+au) (1+bv)}{uv}} \\& \hphantom{{\frac{\partial V_{2}(x,t)}{\partial t}}=}+{\frac{w-w_{\tau_{1}}}{w_{1}}}+\ln{\frac{w_{\tau_{1}}}{w}}. \end{aligned}$$
By using the equilibrium equation, we get \(\lambda=du_{1}+e^{m\tau _{1}}hw_{1}\), \({\frac{h\mu e^{n\tau_{2}}}{k}}={\frac{hw_{1}}{v_{1}}}\) and \({\frac{\beta u_{1}v_{1}}{(1+au_{1})(1+bv_{1})}}=hw _{1}e^{m\tau_{1}}\). Thus, we have
$$\begin{aligned}& {\frac{\partial V_{1}(x,t)}{\partial t}}+hw_{1}{\frac{\partial V_{2}(x,t)}{\partial t}} \\& \quad=e^{-m\tau_{1}}{\frac{\partial u}{\partial t}}+{\frac{\partial w}{\partial t}}+ { \frac{he^{n\tau_{2}}}{k}} {\frac{\partial v}{\partial t}}+{\frac {p}{c}} { \frac{\partial z}{\partial t}}+hw _{1}{\frac{\partial V_{2}(x,t)}{\partial t}} \\& \quad\quad{}-e^{-m\tau_{1}}{\frac{1+au}{1+au_{1}}} {\frac{u_{1}}{u}} { \frac{\partial u}{\partial t}} -{\frac{w_{1}}{w}} {\frac{\partial w}{\partial t}} -{\frac{he^{n\tau_{2}}}{k}} {\frac{v_{1}}{v}} {\frac{\partial v}{\partial t}} \\& \quad= e^{-m\tau_{1}}(du_{1}-du)+hw_{1}-{ \frac{h\mu e^{n\tau _{2}}}{k}}v-{\frac{pq}{c}}z \\& \quad\quad{} +hw_{1}\ln{\frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau _{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}} \\& \quad\quad{} +hw_{1}\ln{\frac{w_{\tau_{1}}}{w}}+ {\frac{he^{n\tau_{2}}}{k}}D \Delta v -e^{-m \tau_{1}}{\frac{u_{1}}{u}} {\frac{1+au}{1+au_{1}}}du_{1} \\& \quad\quad{} +e^{-m\tau_{1}} {\frac{1+au}{1+au_{1}}}du_{1} -{ \frac{u_{1}}{u}} {\frac{1+au}{1+au_{1}}}hw _{1}+ {\frac{v}{v_{1}}} { \frac{1+bv_{1}}{1+bv}}hw_{1} \\& \quad\quad{}+hw_{1} \biggl(1-{\frac{w_{1}}{w}} { \frac{u_{\tau_{1}}}{u_{1}}} {\frac{v_{\tau_{1}}}{v_{1}}} {\frac{(1+au_{1}) (1+bv_{1})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \biggr) \\& \quad\quad{} +pw_{1}z +hw_{1}\biggl(1-{\frac{w_{\tau_{2}}}{w_{1}}} {\frac {v_{1}}{v}}\biggr)-{\frac{he^{n\tau_{2}}}{k}} {\frac{v_{1}}{v}} D \Delta v \\& \quad= -{\frac{de^{-m\tau _{1}}(u-u_{1})^{2}}{u(1+au_{1})}}+hw_{1}\biggl(-1-{\frac{v}{v_{1}}}+ { \frac {v}{v_{1}}} {\frac{1+bv_{1}}{1+bv}}+{\frac{1+bv}{1+bv_{1}}}\biggr) \\& \quad\quad{} +hw_{1} \biggl(4-{\frac{w_{1}}{w}} { \frac{u_{\tau_{1}}}{u_{1}}} {\frac{v_{\tau_{1}}}{v_{1}}} {\frac{(1+au_{1}) (1+bv_{1})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \\& \quad\quad{}-{\frac{u_{1}}{u}} {\frac{1+au}{1+au_{1}}} - {\frac{w_{\tau_{2}}}{w_{1}}} { \frac{v_{1}}{v}}-{\frac{1+bv}{1+bv_{1}}} \biggr)+pz\biggl(w_{1}-{ \frac{q}{c}}\biggr) \\& \quad\quad{} +hw_{1}\ln{\frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau _{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}} { \frac{w_{\tau_{1}}}{w}} \\& \quad\quad{}+{\frac{he^{n\tau_{2}}}{k}}D\Delta v-{\frac{he^{n\tau_{2}}}{k}} { \frac{v_{1}}{v}}D \Delta v \\& \quad= -{\frac{de^{-m\tau_{1}}}{u(1+au_{1})}}(u-u_{1})^{2}- { \frac{bhw_{1}}{v_{1}(1+bv) (1+bv_{1})}} (v-v_{1})^{2} \\& \quad\quad{}+hw_{1} \biggl(4-{\frac{w_{1}}{w}} { \frac{u_{\tau_{1}}}{u_{1}}} {\frac{v_{\tau_{1}}}{v_{1}}} {\frac{(1+au_{1}) (1+bv_{1})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \\& \quad\quad{}-{\frac{u_{1}}{u}} {\frac{1+au}{1+au_{1}}} - {\frac{w_{\tau_{2}}}{w_{1}}} { \frac{v_{1}}{v}}-{\frac{1+bv}{1+bv_{1}}} \biggr)+pz\biggl(w_{1}-{ \frac{q}{c}}\biggr) \\& \quad\quad{} +hw_{1}\ln{\frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau _{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}} { \frac{w_{\tau_{1}}}{w}} \\& \quad\quad{}+{\frac{he^{n\tau_{2}}}{k}}D\Delta v-{\frac{he^{n\tau_{2}}}{k}} { \frac{v_{1}}{v}}D \Delta v \end{aligned}$$
due to
$$\begin{aligned}& \ln{\frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}} {\frac{w_{\tau_{1}}}{w}} \\& \quad= \ln{\frac{u_{1}}{u}} {\frac{1+au}{1+au_{1}}}+\ln {\frac{w_{\tau_{1}}}{w_{1}}} { \frac{v_{1}}{v}} +\ln{\frac{1+bv}{1+bv_{1}}} \\& \quad\quad{}+\ln{\frac{w_{1}u_{\tau_{1}}v_{\tau_{1}}}{wu_{1}v_{1}}} {\frac {(1+au_{1}) (1+bv_{1})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}}. \end{aligned}$$
Then we have
$$\begin{aligned}& {\frac{\partial V_{1}(x,t)}{\partial t}}+hw_{1}{\frac{\partial V_{2}(x,t)}{\partial t}} \\& \quad= -{\frac{de^{-m\tau_{1}}}{u(1+au_{1})}}(u-u_{1})^{2}- { \frac{bhw_{1}}{v_{1}(1+bv) (1+bv_{1})}} (v-v_{1})^{2}-hw_{1} \biggl(G \biggl( {\frac{u_{1}(1+au)}{u(1+au_{1})}} \biggr) \\& \quad\quad{}+G\biggl({\frac{w_{\tau_{2}}v_{1}}{w_{1}v}}\biggr)+G\biggl( {\frac{1+bv}{1+bv_{1}}} \biggr) +G \biggl({\frac{w_{1}u_{\tau_{2}}v_{\tau _{1}}(1+au_{1}) (1+bv_{1})}{wu_{1}v_{1}(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \biggr) \biggr) \\& \quad\quad{}+pz\biggl(w_{1}-{\frac{q}{c}}\biggr)+{ \frac{he^{n\tau_{2}}}{k}}D\biggl(1- {\frac{v_{1}}{v}}\biggr)\Delta v. \end{aligned}$$
We know that
$$ \int_{\Omega} \triangle v\,\mathrm{d}x=0,\qquad \int_{\Omega} {\frac{\triangle v}{v}} \,\mathrm{d}x= \int_{\Omega} {\frac{\Vert \nabla v \Vert ^{2}}{v^{2}}}\, \mathrm{d}x. $$
Thus,
$$\begin{aligned} {\frac{dV(t)}{dt}} &= \int_{\Omega} \biggl( {\frac{\partial V_{1}(x,t)}{\partial t}}+hw_{1}{ \frac{\partial V_{2}(x,t)}{\partial t}} \biggr)\, \mathrm{d}x \\ &=- \int_{\Omega} {\frac{de^{-m\tau_{1}}(u-u_{1})^{2}}{u(1+au_{1})}}\,\mathrm{d}x- \int_{\Omega} {\frac{bhw_{1}(v-v_{1})^{2}}{v_{1}(1+bv) (1+bv_{1})}}\,\mathrm{d}x \\ &\quad{}-hw_{1} \int_{\Omega} \biggl(G \biggl({\frac{u_{1}(1+au)}{u(1+au_{1})}} \biggr)+G \biggl({\frac{w_{\tau_{2}}v_{1}}{w_{1}v}}\biggr) +G \biggl({\frac{w_{1}u_{\tau _{2}}v_{\tau_{1}}(1+au_{1}) (1+bv_{1})}{wu_{1}v_{1}(1 +au_{\tau_{1}}) (1+bv_{\tau _{1}})}} \biggr) \\ &\quad{}+G\biggl({\frac{1+bv}{1+bv_{1}}}\biggr) \biggr)\,\mathrm{d}x+ \int_{\Omega}pz\biggl(w_{1} -{\frac{q}{c}}\biggr) \,\mathrm{d}x+ {\frac{hDe^{n\tau_{2}}v_{1}}{k}} \int_{\Omega}{\frac{\Vert \nabla v \Vert ^{2}}{v^{2}}} \,\mathrm{d}x. \end{aligned}$$
Also, \(R_{1}\leq1\) implies that \(w_{1}\leq{\frac{q}{c}}\). Therefore, if \(R_{1}\leq1< R_{0}\), we can obtain \({\frac{dV(t)}{dt}}\leq0\). Obviously, \({\frac{dV(t)}{dt}}= 0\) if and only if \(u=u_{1}\), \(w=w_{1}\), \(v=v_{1}\) and \(z=0\). It follows that the largest invariant set \(\{(u,w,v,z)\in R ^{4}_{+}: \frac{dV(t)}{dt}=0\}\) is the singleton \(E_{1}\). By LaSalle's invariance principle [30], we finally conclude that the equilibrium \(E_{1}\) is globally asymptotically stable when \(R_{1}\leq1< R_{0}\). □
If \(R_{1}>1\), then the interior equilibrium \(E_{2}\) of model (2) is globally asymptotically stable.
Consider a Lyapunov functional \(L(t)\) as follows:
$$ L(t)= \int_{\Omega} \bigl(L_{1}(x,t)+(hw_{2}+pw_{2}z_{2})L_{2}(x,t) \bigr)\,\mathrm{d}x, $$
$$\begin{aligned}& L_{1}(x,t)= e^{-m\tau_{1}} \biggl(u-u_{2}- \int_{u_{2}}^{u} {\frac{1+a\theta}{1+au_{2}}} { \frac{u_{2}}{\theta}} \,\mathrm{d}\theta \biggr) \\& \hphantom{L_{1}(x,t)= } +w_{2}G\biggl({\frac{w}{w_{2}}}\biggr)+ { \frac{e^{n\tau_{2}}v_{2} (h+pz_{2})}{k}}G\biggl({\frac{v}{v_{2}}}\biggr) +{\frac {pz_{2}}{c}}G \biggl({\frac{z}{z_{2}}}\biggr), \\& L_{2}(x,t)= \int_{0}^{\tau_{1}}G \biggl({\frac{e^{-m\tau_{1}}\beta u_{\theta }v_{\theta}}{(hw_{2}+pw_{2}z_{2}) (1+au_{\theta}) (1+bv_{\theta})}} \biggr) \,\mathrm{d}\theta+ \int_{0}^{\tau_{2}}G\biggl({\frac{w_{\theta}}{w_{2}}}\biggr)\, \mathrm{d}\theta. \end{aligned}$$
By calculating the time derivative of \(L_{1}(x,t)\) and \(L_{2}(x,t)\), we have
$$\begin{aligned} {\frac{\partial L_{1}(x,t)}{\partial t}}&=e^{-m\tau_{1}}\biggl(1-{\frac {1+au}{1+au_{2}}} { \frac{u_{2}}{u}}\biggr){\frac{\partial u}{\partial t}} +\biggl(1- {\frac{w_{2}}{w}} \biggr){\frac{\partial w}{\partial t}}+{\frac {(h+pz_{2})e^{n\tau_{2}}}{k}}\biggl(1-{\frac{v_{2}}{v}} \biggr){\frac{\partial v}{\partial t}} \\ &\quad{}+{\frac{p}{c}}\biggl(1-{\frac{z_{2}}{z}}\biggr){ \frac{\partial z}{\partial t}}, \end{aligned}$$
$$\begin{aligned} {\frac{\partial L_{2}(x,t)}{\partial t}} &={\frac{e^{-m\tau_{1}}\beta uv}{(hw_{2}+pw_{2}z_{2}) (1+au) (1+bv)}} + {\frac{w-w_{\tau_{2}}}{w_{2}}}+\ln{ \frac{w_{\tau_{2}}}{w}} \\ &\quad{}-{\frac{e^{-m\tau_{1}}\beta u_{\tau_{1}}v_{\tau _{1}}}{(hw_{2}+pw_{2}z_{2}) (1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \\ &\quad{}+\ln{\frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}}. \end{aligned}$$
Note that the interior equilibrium \(E_{2}(u_{2}, w_{2}, v_{2}, z_{2})\) satisfies the following equations:
$$ \begin{aligned} & \lambda-du_{2}-{\frac{\beta u_{2}v_{2}}{(1+au_{2}) (1+bv_{2})}}=0, \\ &e^{-m\tau_{1}}{\frac{\beta u_{2}v_{2}}{(1+au_{2}) (1+bv_{2})}}-hw_{2}-pw_{2}z_{2}=0, \\ &ke ^{-n\tau_{2}}w_{2}-\mu v_{2}=0, \\ & cw_{2}z_{2}-qz_{2}=0. \end{aligned} $$
We obtain that \(\lambda=du_{2}+e^{m\tau_{1}}(hw_{2}+pw_{2}z_{2})\), \({\frac{pw_{2}}{v_{2}}}={\frac{p\mu}{k}}\), \(w_{2}={\frac{q}{c}}\) and \(e^{m\tau_{1}}(hw _{2}+pw_{2}z_{2})={\frac{\beta u_{2}v_{2}}{(1+au_{2})(1+bv_{2})}}\). Thus, we have
$$\begin{aligned}& {\frac{\partial L_{1}(x,t)}{\partial t}}+(hw_{2}+pw_{2}z_{2}) { \frac{\partial L_{2}(x,t)}{\partial t}} \\& \quad= e^{-m\tau_{1}}{\frac{\partial u}{\partial t}}+{\frac {\partial w}{\partial t}} +{ \frac{(h+pz_{2})e^{n\tau_{2}}}{k}} {\frac{\partial v}{\partial t}}+{\frac{p}{c}} { \frac{\partial z}{\partial t}} \\& \quad\quad{}+(hw_{2}+pw_{2}z_{2}){ \frac{\partial L_{2}(x,t)}{\partial t}} -e^{-m\tau _{1}}{\frac{1+au}{1+au_{2}}} {\frac{u_{2}}{u}} { \frac{\partial u}{\partial t}} \\& \quad\quad{}-{\frac{w_{2}}{w}} {\frac{\partial w}{\partial t}} - {\frac{(h+pz_{2})e^{n\tau_{2}}}{k}} { \frac{v_{2}}{v}} {\frac{\partial v}{\partial t}} -{\frac{p}{c}} { \frac{z_{2}}{z}} {\frac{\partial z}{\partial t}} \\& \quad= e^{-m\tau_{1}}(du_{2}-du)+hw_{2}+pw_{2}z_{2}+pz_{2}w_{\tau_{2}} -(hw_{2}+pw_{2}z_{2}){\frac{v}{v_{2}}} \\& \quad\quad{} +(hw_{2}+pw_{2}z_{2})\ln {\frac{u_{\tau_{1}} v_{\tau_{1}}}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}} \\& \quad\quad{} +(hw_{2}+pw_{2}z_{2}) {\frac{w_{\tau_{2}}}{w}}+{\frac{e^{n\tau_{2}}(h+pz_{2})}{k}}D \Delta v \\& \quad\quad{}-{\frac{u_{2}}{u}} {\frac {1+au}{1+au_{2}}}(hw_{2}+pw_{2}z_{2})+(hw_{2}+pw_{2}z _{2}){\frac{v}{v_{2}}} {\frac{1+bv_{2}}{1+bv}} \\& \quad\quad{} -e^{-m\tau_{1}}{\frac{u_{2}}{u}} {\frac {1+au}{1+au_{2}}}du_{2}+e^{-m\tau_{1}}{ \frac{1+au}{1+au_{2}}}du _{2} \\& \quad\quad{} -(hw_{2}+pw_{2}z_{2}) { \frac{w_{2}}{w}} {\frac{u_{\tau_{1}}}{u_{2}}} {\frac{v_{\tau_{1}}}{v_{2}}} { \frac{(1+au_{2}) (1+bv_{2})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \\& \quad\quad{}+hw_{2}-(hw_{2}+pw_{2}z_{2}) {\frac{w_{\tau_{2}}}{w_{2}}} {\frac{v_{2}}{v}}+hw_{2}+pw _{2}z_{2} \\& \quad\quad{}-{\frac{e^{n\tau_{2}}(h+pz_{2})}{k}} {\frac{v_{2}}{v}}D\Delta v -pwz_{2}+pw_{2}z _{2} \\& \quad= -{\frac{de^{-m\tau_{1}}(u-u_{2})^{2}}{u(1+au_{2})}}+(hw_{2}+pw_{2}z_{2}) \biggl(-1-{\frac{v}{v_{2}}}+ {\frac{v}{v_{2}}} {\frac{1+bv_{2}}{1+bv}} \\& \quad\quad{} +{\frac{1+bv}{1+bv_{2}}}\biggr) +(hw_{2}+pw_{2}z_{2}) \biggl(4-{\frac{u_{2}}{u}} {\frac{1+au}{1+au_{2}}} - {\frac{w_{\tau_{2}}}{w_{2}}} { \frac{v_{2}}{v}} \\& \quad\quad{}-{\frac{1+bv}{1+bv_{2}}} -{\frac{w_{2}}{w}} {\frac{u_{\tau_{1}}}{u_{2}}} {\frac{v_{\tau_{1}}}{v_{2}}} {\frac{(1+au_{2}) (1+bv_{2})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \biggr) \\& \quad\quad{}+(hw_{2}+pw_{2}z_{2})\ln { \frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}} {\frac{w_{\tau_{2}}}{w}} \\& \quad\quad{}+{\frac{(h+pz_{2})e^{n\tau_{2}}}{k}}\biggl(1- {\frac{v_{2}}{v}}\biggr)D \Delta v \\& \quad= -{\frac{de^{-m\tau_{1}}}{u(1+au_{2})}}(u-u_{2})^{2}- { \frac{b(hw_{2}+pw_{2}z_{2})}{v_{2}(1+bv) (1+bv_{2})}} (v-v_{2})^{2} \\& \quad\quad{}+(hw_{2}+pw_{2}z_{2}) \biggl(4- { \frac{u_{2}}{u}} {\frac{1+au}{1+au_{2}}} -{\frac{w_{\tau _{2}}}{w_{2}}} { \frac{v_{2}}{v}}-{\frac{1+bv}{1+bv_{2}}} \\& \quad\quad{}-{\frac{w_{2}}{w}} {\frac{u_{\tau_{1}}}{u_{2}}} {\frac{v_{\tau_{1}}}{v_{2}}} { \frac{(1+au_{2}) (1+bv_{2})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \biggr) \\& \quad\quad{}+(hw_{2}+pw_{2}z_{2})\ln { \frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}} {\frac{w_{\tau_{2}}}{w}} \\& \quad\quad{} +{\frac{e^{n\tau_{2}}(h+pz_{2})}{k}}\biggl(1- {\frac{v_{2}}{v}}\biggr)D\Delta v \\& \quad= -{\frac{de^{-m\tau_{1}}}{u(1+au_{2})}}(u-u_{2})^{2}- { \frac{b(hw_{2}+pw_{2}z_{2})}{v_{2}(1+bv) (1+bv_{2})}} (v-v_{2})^{2} \\& \quad\quad{}-(hw_{2}+pw_{2}z_{2}) \biggl(G \biggl( {\frac{u_{2}}{u}} {\frac{1+au}{1+au_{2}}}\biggr)+G\biggl({ \frac{w_{\tau _{2}}}{w_{2}}} {\frac{v_{2}}{v}}\biggr)+G\biggl({\frac{1+bv}{1+bv_{2}}} \biggr) \\& \quad\quad{}+G \biggl({\frac{w_{2}}{w}} {\frac{u_{\tau_{1}}}{u_{2}}} { \frac {v_{\tau_{1}}}{v_{2}}} {\frac{(1+au_{2}) (1+bv_{2})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \biggr) \biggr)+ {\frac{e^{n\tau_{2}}(h+pz_{2})}{k}} \biggl(1-{\frac{v_{2}}{v}}\biggr)D\Delta v. \end{aligned}$$
$$ \int_{\Omega}\triangle v\,\mathrm{d}x=0,\qquad \int_{\Omega} {\frac{\triangle v}{v}} \,\mathrm{d}x= \int_{\Omega} {\frac{\Vert \nabla v \Vert ^{2}}{v^{2}}}\, \mathrm{d}x, $$
we further have
$$\begin{aligned} {\frac{dL(t)}{dt}}&= \int_{\Omega} \biggl( {\frac{\partial L_{1}(x,t)}{\partial t}}+(hw_{2}+pw_{2}z _{2}){\frac{\partial L_{2}(x,t)}{\partial t}} \biggr)\,\mathrm{d}x \\ &=- \int_{\Omega} {\frac{de^{-m\tau_{1}}(u-u_{2})^{2}}{u(1+au_{2})}}\,\mathrm{d}x- \int_{\Omega} {\frac{b(hw_{2}+pw_{2}z_{2}) (v-v_{2})^{2}}{v_{2}(1+bv) (1+bv_{2})}} \, \mathrm{d}x \\ &\quad{}-(hw_{2}+pw_{2}z_{2}) \biggl(G\biggl( { \frac{u_{2}}{u}} {\frac{1+au}{1+au_{2}}}\biggr)+G\biggl({\frac{w_{\tau _{2}}}{w_{2}}} { \frac{v_{2}}{v}}\biggr)+G\biggl({\frac{1+bv}{1+bv_{2}}}\biggr) \\ &\quad{}+G \biggl({\frac{w_{2}}{w}} {\frac{u_{\tau_{1}}}{u_{2}}} { \frac{v_{\tau_{1}}}{v_{2}}} {\frac{(1+au_{2}) (1+bv_{2})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \biggr) \biggr) \\ &\quad{}-{\frac{(h+pz_{2})De^{n\tau_{2}}v_{2}}{k}} \int_{\Omega} {\frac{\Vert \nabla v \Vert ^{2}}{v^{2}}}\,\mathrm{d}x. \end{aligned}$$
Therefore, we have \({\frac{dL(t)}{dt}}\leq0\) when \(R_{1}>1\). \({\frac{dL(t)}{dt}}=0\) if and only if \(u=u_{2}\), \(w=w_{2}\), \(v=v_{2}\) and \(z=z_{2}\). It follows that the largest invariant set \(\{(u,w,v,z) \in R^{4}_{+}: \frac{dL(t)}{dt}=0\}\) is the singleton \(E_{2}\). Based on LaSalle's invariance principle [30], we conclude that the equilibrium \(E_{2}\) is globally asymptotically stable when \(R_{1}>1\). □
In the previous section, we analyzed the global stability of a model of virus infection with diffusion and time delay. The object of this section is to further illustrate the obtained theoretical results by some numerical simulations. The Runge-Kutta scheme of fourth order is applied for the reaction part and an explicit Euler scheme for the diffusion part of the PDE. Next, we consider model (2) with the homogeneous Neumann boundary conditions
$$ \frac{\partial v}{\partial\vec{n}}=0, \quad t>0, x=0, 1, $$
and the initial conditions
$$ \begin{aligned} &u(x,\theta)=\phi_{1}(x, \theta)\geq 0, \qquad w(x,\theta)=\phi_{2}(x, \theta)\geq0, \\ &v(x,\theta)=\phi_{3}(x,\theta)\geq0, \qquad z(x,\theta)= \phi_{4}(x, \theta)\geq0, \quad x\in[0,1], \theta\in[-\tau,0]. \end{aligned} $$
In model (2), we choose β, c, q, \(\tau_{1}\), \(\tau_{2}\) as free parameters, and all remaining parameters are fixed as in Table 1.
Table 1 List of parameters
In Figures 1, 2 and 3, (a), (b), (c) and (d) denote time-series figures of \(u(x,t)\), \(w(x,t)\), \(v(x,t)\) and \(z(x,t)\).
Taking \(\pmb{\beta=0.01}\) , \(\pmb{c=0.1}\) , \(\pmb{q=0.12}\) , \(\pmb{\tau _{1}=10}\) , \(\pmb{\tau_{2}=6}\) , we have \(\pmb{R_{0}=0.2066<1}\) , the infection-free equilibrium \(\pmb{E_{0}(1000,0,0,0)}\) is globally asymptotically stable.
Taking \(\pmb{\beta=0.2}\) , \(\pmb{c=0.01}\) , \(\pmb{q=0.2}\) , \(\pmb{\tau _{1}=5}\) , \(\pmb{\tau_{2}=15}\) , we have \(\pmb{R_{0}=3.9696>1}\) , \(\pmb{R_{1}=0.9227<1}\) , the immune-free equilibrium \(\pmb{E_{1}(30.5250,18.4547,2.1179,0)}\) is globally asymptotically stable.
Taking \(\pmb{\beta=0.3}\) , \(\pmb{c=0.01}\) , \(\pmb{q=0.15}\) , \(\pmb{\tau _{1}=10}\) , \(\pmb{\tau_{2}=5}\) , we have \(\pmb{R_{0}=6.2597>1}\) , \(\pmb{R_{1}=1.2303>1}\) , the infection equilibrium only with CTL immune response \(\pmb{E_{2}(10.9402,15,1.7214,0.1272)}\) is globally asymptotically stable.
In this paper, we propose a delayed virus infection model with diffusion, CTL immune responses and Crowley-Martin incidence rate. We have showed the global asymptotic stability of infection-free equilibrium \(E_{0}\), immune-free equilibrium \(E_{1}\) and interior equilibrium \(E_{2}\). By the analysis, the infection-free equilibrium \(E_{0}\) is globally asymptotically stable if \(R_{0}\leq1\); in such circumstances, the viruses are cleared and the infection dies out. If \(R_{1}\leq1< R_{0}\), the immune-free equilibrium \(E_{1}\) is globally asymptotically stable, it is shown that immune response would not be activated and virus infection becomes vanished. If \(R_{1}>1\), model (2) has an interior equilibrium \(E_{2}\), which is globally asymptotically stable. Mathematically, in order to find treatment strategies, we have to find the analytical conditions under which trajectories of the model will converge to infection-free equilibrium. This condition is given in Theorem 3.1, that is, the infection-free equilibrium is globally asymptotically stable if \(R_{0}=\frac{\beta k \lambda}{\mu h(d+a \lambda) e^{m\tau_{1}+n\tau_{2}}}<1\). This relation relies on parameters β, \(\tau_{1}\) and \(\tau_{2}\). That is to say, the variation of these three parameters plays a vital role in disease treatment, and doctors have to adjust these parameters according to the above relation in order to eliminate virus infection. The parameters β, \(\tau_{1}\) and \(\tau_{2}\) might be changed as a consequence of disease treatment. The above analysis shows that \(R_{0}\) increases proportionally to parameter β and decreases proportionally to parameters \(\tau_{1}\) and \(\tau_{2}\). Thus, in order to eliminate the infection, we need to try to increase the value of \(\tau_{1}\) or \(\tau_{2}\). By decreasing infection rate β, immune-free and interior equilibria disappear and infection-free equilibrium will become stable, which implies the virus eradication and that the patient is cured.
Another important issue from medical point of view is to investigate the factors which cause growth of virus. Mathematically, the relations for existence and stability of interior equilibrium need to be investigated. This matter is solved in Theorem 3.3, which gives possible conditions for the existence of interior equilibrium and describes criteria for the stability of interior equilibrium.
Mathematical models can help physicians to choose a suitable dosage and check the effects of their therapeutic action. Virus infection progression has many variations from patient to patient, which is difficult to obtain by an ordinary differential equation. By choosing a relevant diffusive coefficient, the partial differential equation can be varied to best fit the real data according to the progression of different patients. Consequently, the clinician can recommend administration of drugs or treatment strategies to each individual by using the information from the delayed model with the relevant diffusive coefficient.
Our model considers a four-dimensional diffusive virus infection model with intracellular delay, virus replication delay and Crowley-Martin infection rate. It is considered whether the results also can be extended to five-dimensional diffusive virus infection model with mitosis transmission and immune delay. We leave this for the future work.
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This work was supported by 2016xgy041812, NSFXJ (No. 2015211B005), NSFC (No. 11661077), NSFC (No. 11301452), the Youth Cultivation Project for Innovative Talents of Xinjiang (No. qn2015bs013).
Department of Mathematics, Xinjiang Institute of Engineering, Urumqi, Xinjiang, 830091, P.R. China
Chengjun Kang
, Xing Chen
, Jiabo Xu
& Da Huang
College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, 030006, P.R. China
& Hui Miao
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Correspondence to Xing Chen.
KC and MH provided the subject, wrote the introduction and the conclusion and verified some calculations. CX and XJ conceived the study and computed the equilibria and their global stabilities. HD checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
Kang, C., Miao, H., Chen, X. et al. Global stability of a diffusive and delayed virus dynamics model with Crowley-Martin incidence function and CTL immune response. Adv Differ Equ 2017, 324 (2017) doi:10.1186/s13662-017-1332-x
virus infection model
Lyapunov functional
Advances in Fractional Differential Equations and Their Real World Applications


text:
First Choom-Branded Store to Open in Niagara Falls
Vancouver, BC – April 29, 2019 – Choom™ (CSE: CHOO; OTCQB: CHOOF) is pleased to announce that on Friday, April 26, 2019, the Alcohol and Gaming Commission of Ontario ("AGCO") issued a Retail Operator Licence to Lisa Bigioni, and issued a Retail Store Authorization for her Choom-branded cannabis store in Niagara Falls, Ontario, which is expected to open in May 2019.
In accordance with the AGCO's requirements, Ms. Bigioni, as a licensed operator, may finalize her store set-up, including ordering and receiving cannabis product and cannabis accessories into her store ahead of opening for business. The AGCO will also conduct a pre-opening inspection to ensure the store is ready to open for business and sell cannabis products to the public in accordance with all applicable regulations.
Choom and Ms. Bigioni have entered into various definitive agreements, including a licence agreement under which Choom provides its trademarks and branding for the retail store, and an option agreement pursuant to which Choom has the option to acquire, subject to the approval of the AGCO, the retail store after the completion of the lottery process, which expires in December 2019.
"This is a very exciting time for Choom as we help Lisa to open her store in Niagara Falls using the Choom brand." states Chris Bogart, President & CEO of Choom Holdings Inc. "With our expertise in the cannabis retail channels, we are delighted to work with Lisa as she looks forward to opening and operating her Choom-branded store in Niagara Falls. The Niagara region represents a tremendous opportunity for her cannabis retail store, with well over 400,000 people living in the regional municipality and welcoming over 14,000,000 visitors a year."
Lisa Bigioni states "I'm excited to serve the Niagara community with expertise from Choom. Their extensive work in regulated retail markets lends to deep expertise in store setup and operation, but more importantly, brings a focus on education and a dedication to serving the community that I look forward to bringing to my store. I'm excited to welcome everyone to our Choom Niagara store in May."
SAY HELLO TO CHOOM™ Choom™ is an emerging adult use cannabis company whose mission is to establish one of the largest retail networks in Canada and the United States. The Choom brand is inspired by Hawaii's "Choom Gang"—a group of buddies in Honolulu during the 1970's who loved to smoke weed—or as the locals called it, "Choom". Evoking the spirit of the original Choom Gang, our brand caters to the Canadian adult use market with the ethos of 'cultivating good times'. Choom™ is focused on delivering an elevated customer experience through our curated retail environments, offering a diversity of brands for Canadians across a national retail network.
"Chris Bogart" President & CEO
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This section describes how to configure Kerberos for strong authentication for HDP components in an Ambari-managed cluster.
This section describes how to configure Kafka for Kerberos security on an Ambari-managed cluster.
This section describes how to configure Storm for Kerberos security on an Ambari-managed cluster.
HBase is a popular distributed key-value store modeled after Google's BigTable. HBase can support extremely fast lookups, high write throughput and strong consistency making it suitable for a wide variety of use cases ranging from application data store like Facebook Messaging to analytical use cases like Yahoo Flurry. HBase stores data on HDFS, providing it linear scaling as well as fault-tolerance.


text:
Weinmannia richii est une espèce de plantes du genre Weinmannia de la famille des Cunoniaceae.

Synonymes
Weinmannia richii a pour synonymes :
Weinmannia rhodogyne Gibbs
Windmannia richii (A.Gray) Kuntze

Notes et références

Références biologiques

Richii
Espèce de plantes (nom scientifique)


OneRooftops vacation rental website templates come with a web page creation tool. Owners and managers are encouraged to use pages for information that guests will use often, such as an About Us page, an FAQ page or a Contact Us page.
Your website comes with a few default pages already made for you!
Edit or delete a page by clicking the green pencil icon.
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text:
#ifndef _IRTKRICIANNOISE_H

#define _IRTKRICIANNOISE_H

/**
 * Class for adding Rician noise to images
 *
 * This class implements a filter for adding Rician noise to images. The
 * Rician distribution is approiximately Gaussian for high intensity
 * signals, and Rayleigh distributed for low intensities. The Rayleigh
 * intensity distribution can be expressed as:
 * $P_M(M)=\frac{m}{\sigma^2}exp\left(\frac{-M^2}{2\sigma^2}\right)
 * where $M$ is the actual intensity and $\sigma$ is the standard
 * deviation of Gaussian noise. For more information, see Holden et al.,
 * IEEE-TMI 19(2) 2000.
 *
 */

template <class VoxelType> class irtkRicianNoise : public irtkNoise<VoxelType>
{

protected:

  /// Returns the name of the class
  virtual const char *NameOfClass();

public:

  /// Constructor
  irtkRicianNoise(double amplitude = 1);

  /// Destructor (empty).
  ~irtkRicianNoise() {};

  /// Run rician noise filter
  virtual double Run(int, int, int, int);

};

#endif


text:
Maltypus torsae es una especie de coleóptero de la familia Cantharidae.

Distribución geográfica
Habita en Bután.

Referencias

Torsae
Coleópteros del subcontinente indio


text:
package br.ufrn.imd.pitagoras.geneticz.dataStructures;

/**
 * Iterador para um nó binario de valor Double.
 * É utilizado o traversal In-Ordem.
 *
 * @author pitagoras
 * @version 1.5
 */
public class BinTreeIteratorDouble
{

    private BinNodeDouble actualNode, nextNode;

    /**
     * Construtor para objetos da classe BinTreeIterator
     * @param startNode Nó inicial a partir do qual o traversal será iniciado.
     */
    public BinTreeIteratorDouble(BinNodeDouble startNode) throws EmptyTreeException
    {
        // inicializa variáveis de instância
        actualNode = new BinNodeDouble(0, 0.0);

        if(startNode == null){
            throw new EmptyTreeException("O iterador não pode começar a partir de um nó nulo.");
        }
        nextNode = startNode;
        while (nextNode.getLeftSon() != null){
            nextNode = nextNode.getLeftSon();
        }

        try{
            actualNode = iterateNode();
        }catch(Exception e){
            actualNode = nextNode;
            nextNode = null;
        }
    }

    /**
     *
     * @return Se há um elemento atual a ser acessado. Somente é falso quando a iteração tiver acabado.
     */
    public boolean hasActual(){
        return actualNode != null;
    }

    /**
     * @return Se, após o elemento atual, há um elemento para ser acessado.
     */
    public boolean hasNext(){
        return nextNode != null;
    }

    public BinNodeDouble getNode(){
    	return actualNode;
    }

    public double get(){
        return actualNode.getData();
    }

    public int getKey(){
        return actualNode.getKey();
    }

    /**
     * Avança o nó atual do iterador
     */
    public void iterate(){
        actualNode = iterateNode();
    }

    private BinNodeDouble iterateNode(){
        if(nextNode == null){
            return null;
            //throw new EmptyTreeException("Não há um proximo nó na arvore.");
        }

        BinNodeDouble node = nextNode;

        if(nextNode.getRightSon() != null){
            nextNode = nextNode.getRightSon();
            while (nextNode.getLeftSon() != null){
                nextNode = nextNode.getLeftSon();
            }
            return node;
        }else{
            while(true){
                if(nextNode.getFather() == null){
                    nextNode = null;
                    return node;
                }

                if(nextNode.getFather().getLeftSon() == nextNode){
                    nextNode = nextNode.getFather();
                    return node;
                }

                nextNode = nextNode.getFather();
            }
        }
    }
}


A properly manufactured, installed, and maintained floor can last a long time.
Whether you're looking for a soft addition to your family room or a new floor for your outdated bathroom, Flooring Warehouse has the products you need for the project you envision.
Carpet : This lush option provides a cushioned welcome to your feet. Carpet is appropriate for nearly every room of your home. Our variety of color selections is sure to fill your requirements. We have different options for carpet material and thickness, as well. The warm, welcoming appearance of carpet is always inviting.
Vinyl : You might be surprised by the scope of our vinyl flooring selections. If you love the look of hardwood or stone, but would rather not invest in it, we carry vinyl tile options that simulate the effect of real hardwood and stone. Many landlords select vinyl flooring because it is affordable and durable.
Laminate : Your guests may never know your flooring isn't wood! If you are in love with hardwood flooring but not the price, then laminate may be the solution for you. Laminate flooring is almost indistinguishable from true hardwood, and an excellent, cost-effective option that comes in a variety of tones and colors.
Hardwood : Nothing beats the warmth and natural beauty of true hardwood. This elegant option can increase the value of your home and is easy to maintain when properly installed and cared for. With our unbeatable prices, hardwood floors can be within reach.
Tile : Bathrooms and kitchens can benefit from our tiling options. Tile is durable, easy to maintain, and can create rather stunning visual displays. Our tiles are appropriate for shower walls, as well, adding to the versatility of this option. Ceramic and porcelain tiles come in a variety of textures and colors.
Whether you are hoping for a more traditional flooring solution, or if you're searching for something unique and innovative, Flooring Warehouse carries the quality product lines to fulfill your project requests. We can also provide you with professional installation. Call our Austin location at (512) 342-1834, Cedar Park location at (512) 986-7226 , Pflugerville location at (512) 291-2970, or our Georgetown location at (512) 869-7711. Come visit us today!
From design to cleaning, our team can handle all of your flooring needs.
See the amazing work we've done for past clients by viewing our photo gallery.
Get your project started today by filling out our form for a free estimate.


Image Title: Interior White Washed Laminate Wood Floor Idea Inside Whitewash Flooring Decor 4. Post Title: Whitewash Laminate Flooring. Filename: interior-white-washed-laminate-wood-floor-idea-inside-whitewash-flooring-decor-4.jpg. Image Dimension: 500 x 500 pixels. Images Format: jpg/jpeg. Publisher/Author: Javon Conroy. Uploaded Date: Thursday - October 11th. 2018 04:26:10 AM. Category: Architecture. Image Source: lowes.com. White Washed Flooring Floors Love Them Whitewash Laminate Homebase Intended For Remodel 19. Whitewash Oak White Wood Effect Laminate Flooring 3 M Pack With Plan. Innovations Maui Whitewashed Oak 8 Mm Thick X 11 1 2 In Wide 46 Pertaining To Whitewash Laminate Flooring Plan 0. Formica 12mm 1 76sqm White Wash Oak Laminate Flooring Bunnings With Regard To Whitewash Decor 8. Shop Pergo MAX 7 61 In W X 3 96 Ft L Whitewashed Pine Embossed Wood Intended For Whitewash Laminate Flooring Plan 15. Interior White Washed Laminate Wood Floor Idea Inside Whitewash Flooring Decor 4. Laminated Flooring Grey White Washed Laminate Intended For Whitewash Remodel 14. Incredible White Laminate Flooring Home Depot Noble Look Of Any Inside Whitewash Decor 10. Cottage Whitewashed Oak Laminate Flooring Direct Wood Inside Whitewash Remodel 16. Quickstep Reclaim Laminate Flooring 7 48 White Wash Oak Planks With Whitewash Prepare 6.


Unit 7108 is a beautiful ground floor, southern exposure, 2 bedroom/2 bath corner unit only 110 steps from one of the top 10 beaches in the USA, and only steps away from two swimming pools and a courtyard Jacuzzi.
The condo was renovated and redecorated in 2016/2017 and offers all the amenities of home including 3 new wall mounted flat screen HD T.V.'s, wifi, cable, DVD player, land line phone, washer/dryer and wet bar. The master bedroom has carpeting, a queen sized bed, a computer work station and a 38" wall mounted TV. The master bath has a beautiful new marble walk-in shower and floor. The guest bedroom has a double bed, carpeting and a newly renovated guest bathroom. The living room has a new sleeper sofa and leather recliner and has ceramic tile flooring. Sliding plantation shutters are on all doors and windows for added privacy when desired.
This condo offers guests beach living with a homey atmosphere. Welcome to beach and Island living at its best!
Please contact me with information regarding 7108 Jacuzzi Villa 1st Floor.


Going into its third season, the reality show Wicked Tuna needed something a little different so why not Wicked Tuna: North vs South.
Pitting the boats and captains from Gloucester, MA where the show originated, against the finest captains calling Wanchese home, the 2014 version of the popular National Geographic show kicks off tonight (Sunday) at 10 p.m.
The Atlantic bluefin tuna is huge—up to 1200 pounds and over 8' long—powerful and one of the most valuable commercial catches around. Sold mostly on the Japanese sushi market they regularly bring $15,000-$20,000, and occasionally more . . . much more.
But the northern crews did get to the fishing grounds, and even though North Carolina regulations make catching this king of the predator fish more difficult according to reports the competition was fierce and close.


In this section, you will find information about all of Eataly, including the stores and the website.
Search our FAQs - Frequently Asked Questions for answers to your inquiries. If you would like further support, don't hesitate to contact us: click below to send a message or chat with one of our Customer Service representatives (available Monday to Friday from 10 a.m. to 6 p.m.).
For all questions regarding the Stores or Events, you can also directly reach out to the local Guest Relations team. Click here for the contact information.


text:
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using Csla;

namespace InvLib
{
  [Csla.Server.ObjectFactory("InvDal.ProductList,InvDal")]
  public partial class ProductList
  {
    #region Factory Methods

    public static ProductList GetProductList()
    {
      if (_cache == null)
        return DataPortal.Fetch<ProductList>();
      else
        return _cache;
    }

    #endregion
  }
}


Do you wonder how Primal Hair's U.S. patented, clinically proven botanical formula work and how it is different from the two FDA approved drugs, Finasteride and Rogainel, for hair loss?
Let's first start with Finasteride. Finasteride works by blocking an enzyme called 5-alpha reductase, which reduces or blocks the conversion of testosterone to the more potent hormone DHT (dihydrotestosterone). Studies have shown that DHT is closely related to male pattern baldness. Rogaine on the other hand works on widening blood vessels and increasing VEGF (Vascular Endothelial Growth Factor), thus, increasing the density of the hair. However, both drugs also accompanied by side effects such as erectile dysfunction, prostate problem, low libido, hirsutism, skin allergy, etc.
Now, let's look at how PRIMAL HAIR's formula works. PRIMAL HAIR's U.S. patented formula helps reduce hair loss by widening blood vessel and upregulating hair growth factors (VEFG, EGF & FGF-7) expression while decreasing hair loss factors (FGF-5) expression. The more FGF-5 accumulates in our scalp, the more likely that we will suffer from excessive hair loss.
Lastly, you might wonder if taking PRIMAL HAIR has any "side effects". Here are some "side effects" that PRIMAL HAIR users mentioned and we will let you be the judge to decide if you can tolerate them. Potential "side effects" of PRIMAL HAIR include, increased energy level, improved circulation for hands and feet, improved sleeping quality, helping with knee and back soreness, and improved sexual performance*.


During October 2013–June 2014, approximately 54,000 unaccompanied children, mostly from the Central American countries of El Salvador, Guatemala, and Honduras, were identified attempting entry into the United States from Mexico, exceeding numbers reported in previous years (1). Once identified in the United States, U.S. Customs and Border Protection, an agency of the U.S. Department of Homeland Security, processes the unaccompanied children and transfers them to the Office of Refugee Resettlement (ORR), an office of the Administration for Children and Families, U.S. Department of Health and Human Services. ORR cares for the children in shelters until they can be released to a sponsor, typically a parent or relative, who can care for the child while their immigration case is processed. In June 2014, in response to the increased number of unaccompanied children, U.S. Customs and Border Protection expanded operations to accommodate children at a processing center in Nogales, Arizona. ORR, together with the U.S. Department of Defense, opened additional large temporary shelters for the children at Lackland Air Force Base, Texas; U.S. Army Garrison Ft. Sill, Oklahoma; and Naval Base Ventura County, California.
On July 10, 2014, CDC was informed by the California Department of Public Health and ORR about four unaccompanied male children aged 14–16 years with respiratory illnesses at Naval Base Ventura County, three of whom were hospitalized with pneumonia. Among the three patients with pneumonia, two were bacteremic with Streptococcus pneumoniae, ultimately determined to be serotype 5, one of whom also had laboratory-confirmed influenza B virus by polymerase chain reaction (PCR). The fourth patient, without pneumonia, had PCR-confirmed influenza A(H1N1)pdm09. Pneumococcal bacteremia is uncommon among U.S. adolescents, particularly serotype 5, with only three such cases identified in the past 10 years by CDC (2). In addition, influenza activity in the United States is typically lowest in the middle of summer, and Ventura County had no reports of an unusual increase in influenza activity in the community at the time. ORR asked CDC to investigate the scope of this apparent outbreak and implement measures to interrupt transmission.
During July 6–19, 2014, CDC was informed of other clusters of hospitalized children with respiratory disease, increasing the total to 16 cases. The cases were from Naval Base Ventura County (eight cases), Ft. Sill (three), Lackland Air Force Base (two), a standard ORR shelter near Houston, Texas (two), and the Nogales processing center (one). Cases were in persons aged 14–17 years. Diagnoses included laboratory-confirmed pneumococcal pneumonia with laboratory-confirmed influenza (three cases) and without laboratory-confirmed influenza (four cases), influenza pneumonia (one case), and pneumonia with no identified etiology (eight cases). Five patients experienced septic shock requiring intensive care. No case was fatal. All six cases for which pneumococcal isolates were available were identified as serotype 5, a serotype included in 13-valent pneumococcal conjugate vaccine (PCV13) (Prevnar-13, Pfizer). Of the 16 patients identified in this cluster, 11 were tested for influenza viruses; four (36%) were positive (two for influenza A[H1N1]pdm09, one for influenza B, and one for influenza A by rapid test).
Because of the concern that unaccompanied children were at increased risk for influenza and pneumococcal pneumonia in this outbreak setting and the clinically important interaction between influenza and pneumococcal infections (3), CDC recommended that all children residing in temporary or standard ORR shelters receive influenza vaccine and PCV13 in addition to routinely recommended vaccines. Approximately 2,000 children in four affected shelters were vaccinated during July 18–30 with PCV13 and with Food and Drug Administration–approved extended expiration date–specific lots of 2013–14 seasonal influenza vaccine, which includes influenza A(H1N1)pdm09 and influenza B viruses. The shelters reported no serious adverse events.
Although some countries in Central America recommend influenza vaccination for young children, school-aged children generally are not targeted for vaccination (4). Routine annual influenza vaccination is recommended for all persons in the United States aged ≥6 months (5). Because influenza activity was identified among the unaccompanied children, this outbreak underscores the importance of providing routine influenza vaccinations to this population.
PCV13 is routinely given in the United States at age 2–59 months. It is recommended for the older unaccompanied children because of the unexpected number of pneumococcal pneumonia cases occurring in the context of crowded conditions that likely facilitate spread of respiratory agents and because the risk for serious pneumococcal disease is increased with the circulation of influenza viruses.
Efforts by state and local public health departments were crucial in identifying disease clusters among the children, assisting in investigating the clusters, and supporting immunization activities, highlighting the critical role of state and local health departments working with federal agencies in detecting and responding to outbreaks. Additional information about the ongoing humanitarian and public health response is available at http://emergency.cdc.gov/children/unaccompanied/index.asp.
Staff and grantees of the Office of Refugee Resettlement, Administration for Families and Children, US Department of Health and Human Services. Ventura County Public Health, Oxnard; California Department of Public Health. Arizona Department of Health Services. Houston City Health Department, Houston; Harris County Public Health and Environmental Services, Houston; Texas Department of State Health Services.
US Customs and Border Protection. Southwest border unaccompanied alien children. Washington, DC: US Department of Homeland Security, US Customs and Border Protection; 2014. Available at http://www.cbp.gov/newsroom/stats/southwest-border-unaccompanied-children.
CDC. Active Bacterial Core surveillance (ABCs). Atlanta, GA: US Department of Health and Human Services, CDC; 2014. Available at http://www.cdc.gov/abcs/index.html.
Brundage JF. Interactions between influenza and bacterial respiratory pathogens: implications for pandemic preparedness. Lancet Infect Dis 2006;6:303–12.
Ropero-Álvarez AM, Kurtis HJ, Danovaro-Holliday MC, et al. Expansion of seasonal influenza vaccination in the Americas. BMC Public Health 2009;9:361.
CDC. Summary recommendations: prevention and control of influenza with vaccines: recommendations of the Advisory Committee on Immunization Practices (ACIP)—United States, 2013–14. Atlanta, GA: US Department of Health and Human Services, CDC; 2013. Available at http://www.cdc.gov/flu/professionals/acip/2013-summary-recommendations.htm.