text:
Disposable toilet
United States Patent 5524301
A lightweight, portable toilet which is disposable after use. The apparatus is comprised of sturdy biodegradable, environmentally safe corrugated paper board which is capable of being incinerated or buried in the earth's surface after use. The disposable toilet is comprised of a seat member which is placed over the top of a substantially rectangular base member. The seat member contains a rounded rectangular opening. The bottom of the base member is open so as to allow human waste excretions to fall into a pre-excavated hole in the earth's surface located directly underneath the toilet. After use, the toilet can be moved to a new location or disposed of and the hole covered with soil.
Mcguire, David (P.O. Box 129, Apache Junction, AZ, 85217)
MCGUIRE; DAVID
4/449, 4/476
A47K11/02; (IPC1-7): A47K11/02
Field of Search:
4/449, 4/476, 4/483, 4/484
Download PDF 5524301 PDF help
US Patent References:
5048130 Portable, knockdown toilet kit 1991-09-17 Brotman et al. 4/476
3159848 Portable commode 1964-12-08 Arndt 4/476
2912702 Commode 1959-11-17 MacKenzie 4/476
2849726 Portable commode 1958-09-02 Vay 4/476
Phillips, Charles E.
Snell & Wilmer
1. A disposable toilet device comprising:
a base member, comprising at least one die-cut sheet of rigid material, adapted to be folded, thereby defining a planar aspect of said seat member, bordered by a plurality of downwardly projecting sidewalls;
a seat member, comprising at least one die-cut sheet of rigid material, adapted to a be folded, thereby defining a planar aspect of said seat member, bordered by a plurality of downwardly projecting sidewalls;
a central opening defined in said planar aspect of said seat member, said central opening having at least two flaps projecting inwardly from a perimeter of said central opening; and
a slot in the top edge of at least two sidewalls for receiving said at least two projecting flaps of said seat member in order to secure said base member to said seat member, thereby forming a substantially rigid structure.
2. The toilet device of claim 1, wherein said base member further comprises a plurality of substantially rectangular flanges projecting outwardly from said top edge of said base member.
3. The toilet device of claim 2, wherein said base member further comprises a plurality of score lines thereon to facilitate folding said sheet of rigid material to form a substantially rectangular periphery of sidewalls, thereby defining said interior hollow space.
4. The toilet device of claim 1, wherein said seat member further comprises a plurality of substantially rectangular flanges formed on a plurality of edges of said sheet of rigid material, thereby forming said plurality of downwardly projecting sidewalls.
5. The toilet device of claim 4, wherein said seat member further comprises a plurality of score lines thereon, to facilitate folding said sheet of rigid material to form said plurality of downwardly projecting sidewalls.
6. The toilet device of claim 4, wherein said central opening defined in said planar aspect of said seat member is substantially rectangular in form.
7. The toilet device of claim 6, wherein said means for interlocking said seat member to said top edge of said base member further comprises a plurality of reciprocating slots formed in said base member for receipt of said inwardly projecting flaps formed in said central opening of said seat member.
8. The toilet device of claim 1, wherein said rigid material is comprised of corrugated paper board.
9. The toilet device of claim 8, wherein said base member and said seat member are formed by cutting, scoring and bending flat pieces of said corrugated paper board.
10. A method of constructing a disposable toilet device comprising:
cutting out at least one base member and at least one seat member from a flat piece of rigid material, each comprising a substantial rectangular form and having a plurality of rectangular flanges extending from at least one edge of said form;
scoring at least one side of said form to produce a plurality of score lines;
folding said base member along said plurality of score lines to form a substantially rectangular form thereby defining a hollow space;
folding said flanges of said base member, formed along said at least one edge of said rectangular form, along said plurality of score lines, toward said interior hollow space;
cutting a central opening within said seat member such that said central opening includes at least two inwardly projecting flanges extending from a perimeter of said opening;
folding said flanges of said seat member along said plurality of score lines to form a plurality of sidewalls projecting downward from a planar aspect of said seat member;
cutting at least two slots in a top perimeter of said base member to coincide with said at least two inwardly projecting flanges of the seat member;
positioning said seat member, with said sidewalls projecting downward, on said edge of said base member;
interlocking said at least two inwardly projecting flanges of said seat member within said at least two slots of said base member to form said toilet device; and
positioning said toilet device over a pre-excavated hole in the earth's surface.
11. The method of forming a toilet device of claim 10, wherein said step of interlocking said seat member to said base member comprises positioning a plurality of flaps projecting from said seat member into a plurality of reciprocating slots formed on said base member.
The present invention relates to a lightweight, portable toilet for outdoor use which is capable of being disposed after use. The device of the present invention is based on the principle of forming a collapsible rectangular base member and a collapsible seat member and thereafter positioning the member which forms the seat over the top of the rectangular base member to form a disposable toilet. The underneath side of the rectangular base member remains open to allow human waste excretions to fall into a pre-excavated hole in the earth's surface located directly under the toilet. The toilet device of the present inventions construction is collapsed for easy storage and portability to outdoor camping sites. The toilet is composed of a biodegradable, environmentally safe corrugated paper board which allows the entire device to be incinerated when placed into a campfire or buried in the earth's surface.
The conventional toilet, as we know it today, is most commonly found as a permanent structure formed from steel, porcelain or some other sturdy composition. This type of structure is typically found in private, enclosed spaces in homes and businesses. One of the problems associated with this type of permanent structure is the difficulty for the user to carry the device to remote locations because of the weight and size of the device. Remote locations, such as campsites, parks and hiking trails generally do not have their own toilets available. People who enjoy camping at sites far away from established bathrooms cannot enjoy the convenience of a sturdy toilet for support.
In general, the conventional toilet has a central bowl that serves as a receptacle for human waste excretions. The user is supported on a widened frame which is positioned on the outside ring of the bowl. The bowl is filled with water and is permanently affixed above a drain. The water and excretions are ejected into a drain located at the bottom of the bowl through a flushing action. In general, the typical toilet is composed of heavy steel or porcelain which is expensive to purchase and difficult to transport.
In addressing the problem of portability there exists the desire to construct a toilet for use while camping, hiking, hunting or participating in other activities in remote outdoor locations. In doing so, one is faced with looking to the current products of this type on the market. Many of the products on the market today are large permanent structures, such as the previously mentioned steel or porcelain toilet. These type of structures are not suited for outdoor use because the weight of the structures prohibit the user from carrying the device to a remote location. While there exist products on the market which are portable, there remains a need for a toilet which is in addition disposable, collapsible, biodegradable and easy to transport. Many of the now existing portable toilets contain removable containers for collection of the waste. When full, the container must be manually emptied or disposed of. For those types of containers which require manual emptying, there exist an increased risk in the possibility of contact with the excretions and associated bacteria. In addition, many individuals fail to properly dispose of the used container in a waste receptacle, leading to an increase in unsightly and unsanitary campsites.
The device as disclosed is comprised of rigid, lightweight, corrugated paper board which is disposable after use either in a proper waste receptacle, by incineration in a campfire or by burying in the earth's surface. The present invention provides for an interlocking structure assembled from, flat die-cut paper board pieces which are shipped to or purchased flat by the user who at time of assembly folds the pieces along scoring and simply assembles the toilet by interlocking the support members. The device of the present invention is based on the principle of folding the paper board to form the four sides of the collapsible rectangular base member upon which a separately formed seat member is placed. The seat member when placed over the top of the base member forms the disposable toilet of the present invention. The seat member contains a substantially rectangular opening to allow the user to be supported by the side aspects of the seat member, thus being positioned directly over the rectangular opening. The preferred embodiment of the disposable toilet device may be designed to support up to two-hundred and seventy-five pounds of weight dependent upon materials used. The underneath side of the rectangular base member is open to allow for the human waste excretions to fall into a pre-excavated hole in the earth's surface located directly beneath the toilet. After use, the toilet can be alternatively moved to a nearby location for continued use and the hole covered with soil or in the alternative the entire disposable toilet can be incinerated in a campfire or buried in the earth's surface. Thus, the toilet may be moved after each use so there is no accumulation of waste nor the need to continually empty storage containers. The disposable toilet of the present invention is lightweight and capable of being strapped to the average back pack for transportation to campsites, etc. The toilet is comprised of an environmentally safe, biodegradable corrugated paper board and water soluble glue which allows the entire device to be incinerated when placed into a campfire for disposal or buried in the earth's surface. Incineration of the disposable toilet after use provides for an easy and safe sanitary method of disposal.
An alternative embodiment would allow for the positioning of a collection bag, containing sand or similar granular clay-like matter, over the edge of the rectangular base member to enable collection of the human waste excretion. The bag would preferably have a square shaped upper and lower aspect to better accommodate the shape of the disposable toilet and serve the intended purpose.
Accordingly, it is a broad aspect of the present invention to provide for a lightweight, portable toilet which is disposable after use and biodegradable.
It is a further object of the present invention to provide a disposable toilet made of sturdy corrugated paper board to support the weight of various users.
It is a further object of the present invention to provide a biodegradable, environmentally safe disposable toilet.
It is a further object of the present invention to provide a disposable toilet that allows human waste excretions to fall into a pre-excavated hole in the earth's surface located directly below the toilet.
It is a further object of the present invention to provide a disposable toilet that allows human waste excretions to collect in an attached bag.
It is a still further object of the present invention to provide a portable, disposable toilet that can be moved to a nearby location after use without the sanitary problems of accumulation of excretions or the need to empty storage containers.
It is a still further object of the present invention to provide a toilet that can be incinerated in a campfire for sanitary disposal or buried in the earth's surface.
It is a still further object of the present invention to provide a disposable toilet that when collapsed, allows for it to be strapped to the average backpack for transportation to remote locations.
These and other objects, features and advantages of the present invention will become more apparent to those skilled in the art from the following more detailed description of the non-limiting preferred embodiment of the invention taken with reference to the accompanying Figures.
Briefly summarized, a preferred embodiment of the invention is described in conjunction with the illustrative disclosure thereof in the accompanying drawings, in which:
FIG. 1 is a perspective view showing the preferred embodiment of the present invention in its collapsed form;
FIG. 2 is a perspective view showing the preferred embodiment of the partially folded seat member of the present invention;
FIG. 3 is a perspective view showing the preferred embodiment of the partially folded rectangular base member of the present invention;
FIG. 4 is an exploded perspective view showing the completed setup form of the preferred embodiment of the present invention;
FIG. 4A is a partial horizontal cross sectional view taken along line 4A--4A of FIG. 4, showing the corrugation of the paper board of the present invention; and
FIG. 5 is a perspective view showing the placement of the preferred embodiment of the present invention in relation to a pre-excavated hole in the earth's surface.
Referring particularly to the drawings, there is shown the elements of the device of the present invention. In the Figures, there is shown the toilet device 10 of the present invention which comprises a plurality of biodegradable, environmentally safe die-cut corrugated paper board pieces, having rectangular flanges and reciprocating slots, thus capable of frictional engagement so as to construct the disposable toilet 10 of the present invention. Assembly of the disposable toilet does not require the use of tools.
The preferred embodiment of the disposable toilet device 10 as disclosed and shown in FIG. 1 comprises a collapsible rectangular base member 12 and a collapsible seat member 32. The rectangular base member 12 is formed from single die-cut piece of corrugated paper board 14 having a plurality of sides comprising a first short side 16, a second short side (not shown), a first long side 18 and a second long side (not shown). A plurality of rectangular flanges 20 extend outwardly from the top edge of said plurality of sides. The piece of die-cut corrugated paper board 14 which forms the rectangular base member 12 has a series of horizontal score lines 22 located approximately 21/2 inches below the plurality of flange 20 edges. The horizontal scores 22 allow for ease in folding the plurality of flanges 20 at ninety degree angles directed toward the interior aspect of the rectangular base member 12. The rectangular base member 12 contains a plurality of horizontal slots 24, positioned parallel to the top edge and within the first short side 16 and the second short side (not shown). Said horizontal slots 24 are approximately 41/4 inches in length and 1/2 inch in width, and are cut into the rectangular base member 12 along the score line 22, horizontally centered on the first short side 16 and second short side (not shown). A series of vertical scores 26 are formed in the rectangular base member 12 to allow for ease in folding the piece of die-cut corrugated paper board 14 at ninety degree angles to form the vertical sidewalls of the rectangular base member 12. The vertical scores 26, when bent, result in the formation of the rectangular base member 12, having a plurality of formed ninety-degree corners 28, wherein the first short side 16 overlaps 30 the first long side 18 at at least one of said plurality of ninety degree corners 28 to form the rectangular base member 12. A water soluble glue is used to adhere the sides at the point of overlap 30.
The collapsible seat member 32 is formed from a second piece of die-cut corrugated paper board having two short sides 34 and two long sides 36. A plurality of notches 38, approximately 3 inches in length and 5/8 inches in width, are die-cut into each of the two short sides 34 at a distance of 2 inches from each edge of the long sides 36 A 23/4 inch cut 40 extends beyond the proximal end of each notch 38 parallel to the long sides 36. The plurality of notches 38 and cuts 40 formed in the seat member 32 allow for the formation of a plurality of flanges 42 which can be folded along a score line 43 at ninety degree angles parallel to the short sides 34, and perpendicular to the long sides 36. A plurality of rectangular flanges 44 extend outwardly from the edges of the plurality of short sides 34 and long sides 36 to further allow for the formation of the sidewalls of the collapsible seat member 32. A series of horizontal scores 46, located approximately 21/2 inches below the flange 44 edges and parallel to the respective sides, allow for ease in folding the flanges 44 toward the interior aspect of the seat member A plurality of tabs 48, approximately 13/4 inches in length project outwardly approximately 3/4 inch from each short side 34. A plurality of reciprocating slots 50, approximately 2 inches in length and 3/8 inches in width, are located approximately 55/8 inches interior the edge of said short side 34, thus allowing for interlocking of the flanges 48 located on the short sides 34 when folded toward the interior aspect of the seat member 32. A substantially rectangular opening 52, approximately 8 inches in width and 111/2 inches in length, is centrally located within the horizontal plane of the seat member 32. Two projecting flaps 54, approximately 31/2 inches in width and 4 inches in length, protrude outwardly toward the center of the rectangular opening 52 having scored lines formed thereon to permit ease in folding toward the interior aspect of the disposable toilet 10.
The seat member 32 when completely setup rests upon the top edge of the rectangular base member 12. The flaps 54 of the seat member 32 are folded into the interior aspect of the rectangular base member 12 and interlocked into the reciprocating slots 24 of the rectangular base member 12 to rigidly secure the seat member 32 to the base member 12. The disposable toilet 10 is then placed over a pre-excavated hole in the earth's surface.
FIG. 2 shows a perspective view of the preferred embodiment of the partially folded seat member 32 of the present invention. Shown is the seat member 32 with three sides in the completed setup state so as to form three sidewalls perpendicular to the plane of the rectangular opening 52. In forming the fourth sidewall of the seat member 32, the flanges 42 are folded along score lines 43 perpendicular to the long side 36 and positioned parallel to the short side 34. The flange 44 located on the short side 34 is then folded along score lines 46 towards the interior aspect of the seat member 32, over flanges 42. Tabs 48 are interlocked into the reciprocating slots 50 to further stabilize the seat member 32 in its completed setup form.
A perspective view of the preferred embodiment of the partially folded rectangular base member 12 of the present invention in its completed setup form is shown in FIG. 3. The plurality of rectangular flanges 20 are folded along score line 22 toward the interior aspect of the rectangular base member 12. The horizontal slots 24, of the rectangular base member 12, allow for interlocking of flaps 54 of the rectangular opening 52 of the seat member 32 to the rectangular base member 12.
FIG. 4 depicts an exploded perspective view showing the completed setup form of the preferred embodiment of the present invention. Shown is the rectangular base member 12 having rectangular flanges 20 folded toward the interior aspect of the rectangular base member 12. The seat member 32 is placed directly on the rectangular base member 12 and the interlocking flaps 54 of the seat member 32 are interlocked into the horizontal slots 24 of the rectangular base member.
A cross-sectional view taken along line 4A--4A of FIG. 4 shows the double-fluted corrugation composition of the paper board of the present invention.
Placement of the completely setup disposable toilet 10 of the present invention is shown in FIG. 5 in perspective view. The disposable toilet 10 is shown placed directly over a pre-excavated hole 56 to allow for the sanitary deposit of human waste excretions into the earth's surface.
An alternative embodiment of the present invention allows for the incorporation and use of a bag which is positioned over the top edge of the rectangular base member 12 to collect the human waste excretions. The bag would preferably contain sand or similar clay-like matter to act as an absorbent material.
From the above it is apparent that the present invention discloses a portable, disposable toilet assembly which can be easily and simply constructed from a minimum number of parts which interlock to form a rigid toilet structure. The disposable toilet of the present invention is constructed from flat pieces of corrugated paper board to form a disposable toilet that can be shipped flat to a retailer or user who completes the assembly in a few simple operations. The limited number of pieces reduces the weight and cost of producing the disposable toilet, but the particular assemblage of the parts provides the necessary strength along the sides aspects of the seat member to support the user. The disposable toilet may be varied in strength to support loads of varying weights.
From the above description of the preferred embodiment, it is apparent that the present invention provides a disposable, biodegradable toilet which is easily assembled and disposed of by incinerating or burying it in the earth's surface. The initial flat form provides for easy storage, shipping and handling, yet the construction is simple at time of use.
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text:
\section{Introduction}
\noindent
The paper of Kermack and McKendrick (1927)  is one of the best known  contributions to the mathematical  theory of epidemic modelling. The paper  provides the condition of outbreak  and the final size equation in a closed population setting. One of the key features of Kermack and McKendrick (1927) was to introduce an age of infection model. In such a model, the general  infectivity function ($A(\tau)$) of an individual  is considered and depends on the time ($\tau$) elapsed since the infection took place.   Kermack and McKendrick's framework encompasses a wide family of epidemic models; Breda et al (2012) have illustrated the generalisation by providing the following age infection functions for standard SIR and SEIR models.
\begin{equation}
\label{eq:1}
\begin{split}
A(\tau)&=\beta e^{-\alpha \tau} \Longleftrightarrow  SIR \\
A(\tau)&=\beta \frac{\gamma}{\gamma - \alpha}(e^{-\alpha \tau} -e^{-\gamma \tau})\Longleftrightarrow SEIR\\
\end{split}
\end{equation}
\noindent
The paper of Breda et al (2012) \say{On the formulation of epidemic models (an appraisal of Kermack and McKendrick)} revised Kermack and McKendrick's  paper and produced the same results, but the method used was different. In fact, Breda et al (2012) considered  the force of infection as a result of a nonlinear scalar-renewal equation, and they analyzed the cumulative force of infection or the simple force of infection at the disease-free equilibrium and  the endemic equilibrium.\\
\noindent
 In the current paper, we  investigate the effects of an adaptive vaccination strategy on the dynamics of infectious diseases in a closed  population  and a demographically open population. The methodology and  key assumptions are based on Breda et al (2012). \\
\begin{figure}[h]
\vspace{-20pt}
\centering
\includegraphics[width=0.5\textwidth]{diag}
\vspace{-15pt}
\caption{
The transfer diagram of the model: the force of infection function (the probability per unit of time that a susceptible becomes infected)   $t \mapsto F(t)$; the rate of vaccination function $t\mapsto\phi(t)$; $S$
(non-vaccinated susceptible); $V$ (vaccinated susceptible); $\theta$ (vaccine parameter ($0\leq\theta\leq1$)); $I$ (infected population); $B$ (constant birth rate); $\mu$ (constant per capita death rate).}
\vspace{-10pt}
\label{fig:diag}
\end{figure}

\noindent
The epidemic model and the vaccination process are illustrated by Figure \ref{fig:diag}. The susceptible population is  divided into non-vaccinated susceptible and vaccinated susceptible.
The main assumptions are that  the infection leads to permanent immunity (no re-infection); the force of infection occurring in non-vaccinated susceptibles  is proportional to the force of infection  within the vaccinated susceptibles; and the rate of vaccination is proportional to the force of infection in the non-vaccinated susceptibles. In fact, a natural feature of an adaptive vaccination policy is that the rate of vaccination should increase when the force of infection increases and decrease when the force of infection decreases.
Our paper will first analyze the closed population setting, and then introduce a survival function and analyze  an age-structured population.

\section{Closed population epidemic model}
\noindent
The dynamic of infection can be described in each susceptible group as illustrated in the transfer diagram of the model.  The instantaneous change in the susceptibles is determined by the number of new cases of susceptibles infected per unit of time (incidence) and the number of new vaccinated susceptibles per unit of time. In a closed population, the number of susceptibles only changes due to transmission of infection and vaccination; thus, $\mu=B=0$ in Figure \ref{fig:diag}.
The following system of equations can be derived.
\begin{equation}
\label{eq:2}
\begin{split}
\frac{dS(t)}{dt}&= -F(t)S(t)-\phi(t)S(t) ,\\
\frac{dV(t)}{dt}&= -\theta F(t)V(t)+\phi(t)S(t) .
\end{split}
\end{equation}
\noindent
The initial conditions are given by $S(-\infty)>0$ and $V(-\infty)=0$.\\
\noindent
By solving the system of equations (\ref{eq:2}), we have the  following results.
\begin{equation}
\label{eq:6}
\begin{split}
S(t)&\;=\;S(-\infty)e^{-(y+\Phi)(t)} , \\
(S+V)(t)&\;=\;S(-\infty)C(t)e^{-\theta y(t)} ,\\
\end{split}
\end{equation}
where
\begin{equation}
\label{eq:6a}
\begin{split}
 y(t)&\;=\; \int_{-\infty}^{t}F(\sigma)d\sigma,  \quad \Phi(t)\;=\;\int_{-\infty}^{t}\phi(\sigma)d\sigma \,,
  \quad  \hbox{and}
  \\
 C(t)&\;=\; 1-(1-\theta)\int_{-\infty}^{t}F(\sigma)e^{-(1-\theta)y(\sigma)-\Phi(\sigma)}d\sigma \,.  \\
\end{split}
\end{equation}

\noindent
The force of infection depends on the size of the infectious population. The rate of new
infections at time $t-\tau$ is $F(t-\tau)S(t-\tau)$ from the non-vaccinated susceptibles and
$\theta F(t-\tau)V(t-\tau)$ from the vaccinated susceptibles. After  $\tau$ additional units of time, these
cases contribute
$(F(t-\tau)S(t-\tau)+ \theta F(t-\tau)V(t-\tau))A(\tau)$ to the force of infection at time $t$.
By summing all the contributions with respect to the elapsed time $\tau$, we obtain
the scalar-renewal equation
\begin{equation}
\label{eq:3}
F(t)=\int_{0}^{\infty} (F(t-\tau)S(t-\tau)+ \theta F(t-\tau)V(t-\tau))\,A(\tau)\, d\tau .
\end{equation}
\noindent
Taking into account the equations (\ref{eq:3}) and (\ref{eq:2}), we derive the following cumulative force of infection at each time $t$, denoted $y(t)$.
\begin{equation}
\label{eq:4}
\begin{split}
y(t) & =\int_{-\infty}^{t}F(\sigma)d\sigma \\
    &= \int_{0}^{\infty}\int_{-\infty}^{t}-\frac{d(S+V)}{dt}(\sigma-\tau)\,\,A(\tau)\,d\tau \,d\sigma \\
     & =\int_{0}^{\infty} S(-\infty)\left(1-\frac{(S+V)(t-\tau)}{S(-\infty)}\right)\, A(\tau)\,d\tau .
 \end{split}
\end{equation}
\noindent
When $t \rightarrow \infty$, the equation (\ref{eq:4}) becomes
\begin{equation}
\label{eq:5}
\begin{split}
y(\infty)&\;=\;  \left(1-\frac{(S+V)(\infty)}{S(-\infty)}\right)\int_{0}^{\infty} S(-\infty)\,A(\tau)\,d\tau \\
     & =\; R \left(1-\frac{(S+V)(\infty)}{S(-\infty)}\right) \hspace{10mm}
 \hbox{where} \hspace{5mm}
 R\;=\;\int_{0}^{\infty} S(-\infty)\,A(\tau)\,d\tau \,.
 \end{split}
\end{equation}
\noindent
In the expression for $C(t)$ in \eqref{eq:6a}, the integral
$\int_{-\infty}^{t}F(\sigma)e^{-(1-\theta)y(\sigma)-\Phi(\sigma)}d\sigma\ $ is not easy to handle in general,
as the rate of vaccination  ($\phi(t)$) is unknown. We consider a special case by  assuming  a linear relationship between the  vaccination rate  and the force of infection. Specifically, we assume
$\phi(t)=pF(t)$ where $p$ is the vaccination rate parameter. \\
With this assumption on $\phi(t)$, the following results can be derived.
\begin{equation}
\label{eq:7}
\begin{split}
C(\infty)&\;=\;\frac{1}{1+p-\theta}\left(p+(1-\theta)e^{-(1+p-\theta)y(\infty)}\right)\,,  \\
\frac{(S+V)(\infty)}{S(-\infty)} &\;=\;C(\infty)e^{-\theta y(\infty)}\\
                                 &\;=\;  \frac{1}{1+p-\theta}\left(pe^{-\theta y(\infty)}+(1-\theta)e^{-(1+p)y(\infty)}\right) \,.
\end{split}
\end{equation}

\subsection{Impact on the endemic steady state}
\noindent
The expression in \eqref{eq:7} was replaced into the equation  \eqref{eq:5}, which describes the asymptotic behaviour of the cumulative force of infection in  the epidemic dynamic.  The result is
\begin{equation}
\label{eq:8}
y(\infty) \;=\;R\left(1-\frac{pe^{-\theta y(\infty)}+(1-\theta)e^{-(1+p)y(\infty)}}{1+p-\theta} \right)
 \hspace{5mm} \hbox{with}\hspace{5mm}
  R=\int_{0}^{\infty} S(-\infty)\,A(\tau)\,d\tau \,.
\end{equation}
\noindent
According to Breda et al (2012), the reproduction number ($R$) defined in \eqref{eq:8} can be interpreted as the expected number of secondary cases caused by a primary case introduced into a population with susceptible density $S(-\infty)$. The case $R>1$ is the interesting case to study if we want to investigate the effect of the vaccination rate parameter ($p$)
and the vaccine parameter ($\theta$) on the cumulative infection force ($y(\infty)$). \\
\noindent
\\
In order to analyse the equation (\ref{eq:8}) above, we have to know the classical properties of the solution of the equation $x=W(x)e^{W(x)}$, where the solution  W(x) is called the Lambert function.\\
\begin{figure}[h]
\vspace{-5pt}
\centering
\includegraphics[width=0.5\textwidth]{lam}
\vspace{-5pt}
\caption{
Solution of the equation $x=W(x)e^{W(x)}$ }
\vspace{-5pt}
\label{fig:diag2}
\end{figure}

\noindent
As shown in the Figure \ref{fig:diag2}, the solution  $W(x)$ is actually a multivalued function, with two branches denoted $W_0$ and $W_{-1}$  defined as follows:
\begin{center}
   \[W(x)=
  \begin{cases}
  W_0(x)   & \quad \text{if } -\frac{1}{e}\;\leq \;x \,,  \\
W_{-1}(x) & \quad \text{if } -\frac{1}{e}\;\leq \; x \;<\;0 \,.
  \end{cases}
\]
\end{center}

\begin{lemma}
   \label{lem.W}
Let  $R$ be a positive real number, and consider nonnegative solutions $u$ of the equation
\[      u \;=\; R(1-e^{-u}) \,.    \]
(a)  If  $0<R\leq 1$, then $u=0$ is  the only nonnegative solution.
\\
(b)  If  $R>1$, then there are two nonnegative solutions:  $u=0$ and $u=R+W_0(-Re^{-R})$.
\end{lemma}
\noindent
\textbf{Proof:}\\
 Part (a): The function $g(u)=u-R(1-e^{-u})$ is strictly increasing for $0\leq u$. Therefore the unique root  is $u=0$.\\
 Part (b) : We have $u-R=-R{e^{-R}}{e^{-u+R}}$ , which implies $(u-R){e^{u-R}}=-R{e^{-R}}$. By using  the Lambert function, we have $u=R+W(-R{e^{-R}})$.\\
For $R>1$, using the identity property of the Lambert function, we have
\begin{center}
   \[W(-R{e^{-R}})=
  \begin{cases}
  W_0(-R{e^{-R}})  & >\; -1,  \\
W_{-1}(-R{e^{-R}}) & =\; -R.
  \end{cases}
\]
\end{center}
\noindent
Therefore, we have  two solutions $u=0$ and $u=R+W_0(-Re^{-R})$.\\
\medskip

\noindent
\textbf{Case 1: ineffective vaccine($\theta =1$) }\\
 The expression in \eqref{eq:8} becomes
\begin{equation}
\label{eq:9}
y(\infty) =R\left(1-e^{- y(\infty)}\right).\ \
\end{equation}
\noindent
 The quantity $y(\infty)$ is  the same as the cumulative force of infection without vaccination
(Breda et al (2012)).  Using  Lemma \ref{lem.W}, the nonzero solution is $y(\infty)=R+W_0(-Re^{-R})$, because the hypothesis is that $R>1$.\\

\medskip
\noindent
\textbf{Case 2: 100\% effective vaccine ($\theta=0$) } \\
The expression in \eqref{eq:8} becomes
\begin{equation}
\label{eq:10}
y(\infty) =R\left(1-\frac{p+e^{-(1+p)y(\infty)}}{1+p}\right ).
\end{equation}
Using  Lemma \ref{lem.W} with the substitution $u=(1+p)y(\infty)$, we find that Equation (\ref{eq:10})
has the positive solution  $y(\infty) \,=\,  [R+W_0(-Re^{-R})]/(1+p)$ when $R>1$. Thus we approach a disease-free steady state as $p$ gets large.\\

\medskip
\noindent
\textbf{Case 3: $p \rightarrow \infty$ and $\theta\neq0$}\\
 By increasing the vaccination rate parameter ($p$), the expression in \eqref{eq:8} becomes
\begin{equation}
\label{eq:11}
y(\infty) =R\left(1-e^{-\theta y(\infty)}\right).
\end{equation}
Using  Lemma \ref{lem.W} with the substitution $u=\theta y(\infty)$, we find the solution of the equation \eqref{eq:11} to be
\[      y(\infty)\;=\;
  \begin{cases}
  0     & \quad \text{if } \theta\leq \frac{1}{R} \,, \\
R +\frac{1}{\theta}W_0(-\theta R e^{-\theta R})
 & \quad \text{if } \frac{1}{R}< \theta\leq 1\,.
  \end{cases}
\]
 \medskip

 \noindent
\textbf{Case 4: $0<\theta<1$ and $p>0$}\\
The expression in  \eqref{eq:8} can be transformed into the following  standard equation \eqref{eq:12}  with one unknown variable $x$, which is the variable of interest.
 \begin{equation}
\label{eq:12}
\begin{split}
x-R\left(1-\frac{pe^{-\theta x}+(1-\theta)e^{-(1+p)x}}{1+p-\theta}\right) \;=\; 0 \hspace{7mm}
\hbox{where}  \ \
 x=y(\infty)
 \end{split}
\end{equation}
 \noindent
To consider a representative situation, we supposed the reproduction number ($R$) is equal to 2. As illustrated  in the Figure \ref{fig:test1}, the result shows that the effective vaccine is the most determinant  factor.  In fact, when the vaccine is not effective (high vaccine parameter), whatever the vaccination rate parameter chosen, the effect on $y(\infty)$ is marginal (yellow area). On the other side, $y(\infty)$ is responsive to both factors when the vaccine parameter is low; the degree of $y(\infty)$ reduction depends on the vaccination rate parameter. As shown in Figure \ref{fig:test1}, the color on the graph becomes blue quickly when the vaccination rate parameter increases. In Figure \ref{fig:test1}, the graphs \ref{fig:sub11} and \ref{fig:sub12} are two views of the same function in a three-dimensional space, whereas the graph \ref{fig:sub13} is a two-dimensional space.
\begin{figure}[h]
 \vspace{-6pt}
    \begin{subfigure}{.5\linewidth}
\centering
    \includegraphics[scale=0.4]{plot11}
\caption{\tiny Cumulative force of infection $y(\infty)$ as a function of $\theta$ and $p$}
\vspace{10pt}
\label{fig:sub11}
\end{subfigure}%
\begin{subfigure}{.5\linewidth}
\centering
    \includegraphics[scale=0.4]{plot12}
\caption{\tiny Cumulative force of infection $y(\infty)$ as a function of $\theta$ and $p$}
  \vspace{10pt}
\label{fig:sub12}
\end{subfigure}\\[1ex]
\begin{subfigure}{\linewidth}
\centering
 \vspace{-15pt}
    \includegraphics[scale=0.5]{plot133}
 \caption{\tiny  Cumulative force of infection $y(\infty)$ in percentage (vaccination versus non vaccination scenario)}
 \vspace{-10pt}
\label{fig:sub13}
\end{subfigure}
\caption{Impact of adaptive vaccination strategy on cumulative force of infection $y(\infty)$.}
\label{fig:test1}
 \vspace{-10pt}
\end{figure}

\newpage
\section{ Age-structured epidemic model}
\noindent
 In this section, we consider the situation where, at the population level, new susceptibles arise as a result of reproduction at a constant birth rate $B$. In addition, we consider the survival function $\mathscr{F}(a)$, which describes the probability that a newborn individual lives at least until age $a$. If at time $t$ a susceptible has age $a$, then at time $t-a+\sigma$  this susceptible has age $\sigma$
($0<\sigma\leq a$). For a small positive time duration $h$, taking into account the survival function $\mathscr{F}(a)$, the behavior of the non-vaccinated susceptibles $S(t,a)$ at time $t$ and  at age $a$ follows the equation
\begin{equation}
  \label{eq:13}
S(t-a+\sigma+h,\sigma+h)\;=\;S(t-a+\sigma,\sigma)\frac{\mathscr{F}(\sigma+h)}{\mathscr{F}(\sigma)}
  (1-F(t-a+\sigma)h-\phi(t-a+\sigma)h+o(h^2)) \,.
\end{equation}
 By re-arranging, we have the following approximation.
\begin{equation*}
\begin{split}
\frac{\frac{S(t-a+\sigma+h,\sigma+h)}{\mathscr{F}(\sigma+h)} - \frac{S(t-a+\sigma,\sigma)}{\mathscr{F}(\sigma)}}{h}=-F(t-a+\sigma)\frac{S(t-a+\sigma,\sigma)}{\mathscr{F}(\sigma)}-\phi(t-a+\sigma)\frac{S(t-a+\sigma,\sigma)}{\mathscr{F}(\sigma)}+\frac{o(h^2)}{h}.
\end{split}
\end{equation*}
We can use similar reasoning for the vaccinated susceptibles $V(t,a)$ at time $t$ and  at age $a$, and
then take the limit as $h$ converges to 0,  resulting in the system of differential equations
\begin{equation}
\label{eq:16}
\begin{split}
\frac{d(\frac{S(t-a+\sigma,\sigma)}{\mathscr{F} (\sigma)})}{d\sigma} &\;=\;
-F(t-a+\sigma)\frac{S(t-a+\sigma,\sigma)}{\mathscr{F}(\sigma)}\;-\;\phi(t-a+\sigma)\frac{S(t-a+\sigma,\sigma)}{\mathscr{F}(\sigma)}, \\
\frac{d(\frac{V(t-a+\sigma,\sigma)}{\mathscr{F} (\sigma)})}{d\sigma}  &\;=\;
-\theta F(t-a+\sigma)\frac{V(t-a+\sigma,\sigma)}{\mathscr{F}(\sigma)}\;+\;\phi(t-a+\sigma)\frac{S(t-a+\sigma,\sigma)}{\mathscr{F}(\sigma)}.
\end{split}
\end{equation}
\noindent
 If there is no infection in the population, then we have a stable age  distribution (Breda et al (2012)), with
$S(t,a)$ and $V(t,a)$ becoming
\begin{equation*}
\begin{split}
S(t,a)&\;=\;B\mathscr{F} (a) \,, \\
V(t,a)&\;=\;0 \,.
\end{split}
\end{equation*}
\noindent
Recall from the transfer diagram of the model that $B$  is the constant birth rate.\\
More generally, by solving the system of equations \eqref{eq:16}, we obtained the  following results.
\begin{equation}
\label{eq:16a}
\begin{split}
S(t,a)&=B\mathscr{F} (a)e^{-\int_{0}^{a}(F+\phi)(t-a+\sigma)d\sigma}, \\
V(t,a)&=B\mathscr{F} (a)e^{-\theta\int_{0}^{a}F(t-a+\sigma)d\sigma} \int_{0}^{a}\phi(t-a+\sigma)e^{-\int_{0}^{\sigma}((1-\theta)F+\phi)(t-a+\tau)d\tau} d\sigma.
\end{split}
\end{equation}
\noindent
The force of infection depends  on   the size of the infectious population and the survival function characteristics of the population. At time $t$, from individuals who were infected at time $t-\tau$  at age $a$, the contribution to the force of infection is the product of $(F(t-\tau)S(t-\tau,a)+ \theta F(t-\tau)V(t-\tau,a))A(\tau)$ infectious individuals  and a demographic factor $\frac{\mathscr{F} (a+\tau)}{\mathscr{F} (a)}$, which is the fraction of infectious individuals  who survive from age $a$ to age $a+\tau$. By summing all the contributions with respect to to the elapsed time $\tau$ and with respect to the age $a$, we get the following renewal equation.
\begin{equation}
\label{eq:17}
\begin{split}
F(t) &\;=\; \int_{0}^{\infty}\int_{0}^{\infty} (F(t-\tau)S(t-\tau,a)+ \theta F(t-\tau)V(t-\tau,a))A(\tau)\frac{\mathscr{F} (a+\tau)}{\mathscr{F} (a)} \ d\tau \ da  \\
     &\;=\; \int_{0}^{\infty}F(t-\tau) \int_{0}^{\infty} (S(t-\tau,a)+ \theta V(t-\tau,a))\frac{\mathscr{F} (a+\tau)}{\mathscr{F} (a)}A(\tau) \ d\tau \ da \,.
\end{split}
\end{equation}
\noindent
The integral  $\int_{0}^{a}\phi(t-a+\sigma)e^{-\int_{0}^{\sigma}((1-\theta)F+\phi)(t-a+\tau)d\tau} d\sigma$ from \eqref{eq:16a} is not easy to handle  because the rate of vaccination function $(\phi(t))$ is unknown.
As in Section 2, we assume a linear relationship between the  vaccination rate and the force of infection, in
the form $\phi(t)\,=\,pF(t)$  where $p$ is the vaccination rate parameter. The solution \eqref{eq:16a} becomes:
\begin{equation}
\label{eq:19}
\begin{split}
S(t,a)&=B\mathscr{F} (a)e^{-(1+p)\int_{0}^{a}F(t-a+\sigma)d\sigma}  \,,  \\
V(t,a)&=B\mathscr{F} (a)\frac{p}{1+p-\theta}\left(e^{-\theta\int_{0}^{a}F(t-a+\sigma)d\sigma} -e^{-(1+p)\int_{0}^{a}F(t-a+\sigma)d\sigma}\right) \,.
\end{split}
\end{equation}

\subsection{Characteristic equation of the endemic steady state}
\noindent
The solution \eqref{eq:19} can be substituted into the renewal equation \eqref{eq:17}. In the endemic steady state, we assume the force of infection ($F(t)$) converges to a constant value $F$. When $t$ goes to $+\infty$, the renewal equation can be rearranged, leading to the following characteristic equation of the endemic steady state.
\begin{equation}
\label{eq:20}
 1\;=\;\frac{B}{1+p-\theta}\int_{0}^{\infty}\int_{0}^{\infty} \left(p\theta e^{-\theta aF}+(1+p)(1-\theta)e^{-(1+p)aF}
    \right) \mathscr{F} (a+\tau)\,A(\tau) \, d\tau \, da \,.
\end{equation}
\noindent
In order to study the properties of the characteristic equation, we consider the following function:
\begin{equation}
\label{eq:21}
 f(x)\;=\;\frac{B}{1+p-\theta}\int_{0}^{\infty}\int_{0}^{\infty} \left(p\theta e^{-\theta ax}+(1+p)(1-\theta)e^{-(1+p)ax}
 \right) \mathscr{F} (a+\tau)\,A(\tau) \, d\tau \, da \,.
\end{equation}
\noindent
It is obvious that $f(x)$ is decreasing in $x$,   and
$f(0)=B\int_{0}^{\infty}\int_{0}^{\infty}\mathscr{F} (a+\tau)A(\tau) \, d\tau \, da$. The condition $f(0)>1$ is sufficient
to guarantee the existence of a solution $F$ of equation \eqref{eq:20} with $F \neq 0$.
Furthermore, $f(0)$ depends only on the constant birth rate $B$, the survival function $\mathscr{F}$,
and the expected contribution to the force of infection $A$, since
the vaccine parameters $\theta$ and $p$ cancel out.  Indeed, as noted by Breda et al (2012),
$f(0)$ is  the basic reproduction number and can interpreted as the expected number of secondary cases caused by a primary case introduced in a susceptible population with age distribution $B\mathscr{F}(a)$.
\begin{equation}
\label{eq:22}
 R\;=\;f(0)\;=\; B\int_{0}^{\infty}\int_{0}^{\infty}\mathscr{F} (a+\tau)\,A(\tau) \,d\tau \, da \,.
\end{equation}

\subsection{General case}
\noindent
Throughout this subsection, we shall assume $R \,=\, f(0) \, >\,1$.
\noindent
The implicit function theorem can be used on equation \eqref{eq:20}  to express $F$ as a function of two variables $p$ and $\theta$. \\
Let us define
$$G(\theta,p,F)=\frac{B}{1+p-\theta}\int_{0}^{\infty}\int_{0}^{\infty} (p\theta e^{-\theta aF}+(1+p)(1-\theta)e^{-(1+p)aF})\mathscr{F} (a+\tau)A(\tau) \ d\tau \ da -1.$$
Then $G$ is a continuously differentiable function;  $G(\theta,p,F)=0$ if $F$ is the solution of equation \eqref{eq:20}; and $\frac{dG(\theta,p,F)}{dF}<0$.
Therefore $F(\theta,p)$ is a continuously differentiable function and the solution of the following  equation.
\begin{equation}
\label{eq:31}
 1=\frac{B}{1+p-\theta}\int_{0}^{\infty}\int_{0}^{\infty} (p\theta e^{-\theta aF(\theta,p)}+(1+p)(1-\theta)e^{-(1+p)aF(\theta,p)})\mathscr{F} (a+\tau)A(\tau) \ d\tau \ da .\\
\end{equation}
\smallskip

\noindent
\textbf{Case 1: ineffective vaccine ($\theta =1$)}\\
When $\theta=1$, the equation \eqref{eq:31} becomes
\begin{equation}
\label{eq:32}
 1=B\int_{0}^{\infty}\int_{0}^{\infty} e^{-aF(1,p)}\mathscr{F} (a+\tau)A(\tau) \ d\tau \ da .
\end{equation}
\noindent
We have $F(1, p)=F^*$, where $F^*$ comes from the non-vaccination case studied by Breda et al (2012). Therefore,  the endemic force of infection is  the same as the endemic force of infection without vaccination.\\

\noindent
\textbf{Case 2: 100\% effective vaccine ($\theta= 0$)} \\
When $\theta=0$, the equation \eqref{eq:31} becomes
 \begin{equation}
\label{eq:33}
 1=B\int_{0}^{\infty}\int_{0}^{\infty} e^{-(1+p)aF(0,p)}\mathscr{F} (a+\tau)A(\tau) \ d\tau \ da .
\end{equation}
\noindent
Similarly to Case 1, we have $(1+p)F(0,p) =F^*$ for  $p>0$.
Thus  $F(0,p) =\frac{F^*}{p+1}$, which depends on the factor  $\frac{1}{p+1}$, and we see that
$\lim_{p\to\infty}F(0,p)=0$, which corresponds to the disease-free steady state.\\

\noindent
\textbf{Case 3: $p \rightarrow \infty$ and $\theta\neq0$}\\
We assumed  $\lim_{p\to\infty}F(\theta,p)=F(\theta, +\infty)$.\\
We shall show that the solution of the equation \eqref{eq:31} becomes:
\begin{equation}
   \label{eq:30A}
    F(\theta,+\infty)=
  \begin{cases}
  0     &  \text{if } \theta\leq \frac{1}{f(0)} \,, \\
   F(\theta,+\infty) &  \text{if } \frac{1}{f(0)}< \theta<1 \,,\\
 F^*   &  \text{if } \theta=1  \,.
  \end{cases}
\end{equation}
In addition, for $\frac{1}{f(0)}< \theta<1$, $F(\theta,+\infty)$ is an  increasing function and $0< F(\theta,+\infty)<F^*$.\\

\noindent
a) For $\theta=1$ : \\
 By Case 1 above, $F(1, p)=F^* $ for all $p>0$. Therefore,  $F(1,+\infty)=F^*$.
 \\
\noindent
\\
b) For $0<\theta\leq \frac{1}{f(0)}$ :  \\
 We shall assume $F(\theta,+\infty)>0$.\\
Let us take a sequence $(p_n)$ such that $\lim_{n \to \infty}p_n=+\infty$,
so that $\lim_{n \to \infty}F(\theta,p_n)=F(\theta, +\infty)$. \\
 We consider  the sequence of functions $(f_n)$ defined by
\begin{equation*}
f_n(a,\tau)=\frac{B}{1+p_n-\theta} \left(p_n\theta e^{-\theta aF(\theta,p_n)}+(1+p_n)(1-\theta)e^{-(1+p_n)aF(\theta,p_n)}\right)\mathscr{F} (a+\tau)A(\tau) \,.
\end{equation*}
According to the definition of $F(\theta,p_n)$ and the equation  \eqref{eq:31}, we have $\int_{0}^{\infty}\int_{0}^{\infty}f_n(a,\tau) \ d\tau\ da=1$.
The function $f_n$ is dominated by an integrable function $ B\mathscr{F} (a+\tau)A(\tau)$. In fact, we have
 \begin{equation*}
f_n(a,\tau) \;\leq \; B\,\mathscr{F} (a+\tau)A(\tau)
     \hspace{5mm} \hbox{\and}  \hspace{5mm}
f(0) \;=\; \int_{0}^{\infty}\int_{0}^{\infty}B\mathscr{F} (a+\tau)A(\tau)\, d \tau\, da \,.
\end{equation*}
 \noindent
Following the Dominated Convergence Theorem (DCT), and recalling $F(\theta,+\infty)>0$, we have
 \begin{equation}
  \label{eq:30B}
   1\;=\; \int_{0}^{\infty}\int_{0}^{\infty}\lim_{n \to \infty}f_n(a,\tau)\,  da\  d\tau \;=\;
 B\theta\int_{0}^{\infty}\int_{0}^{\infty}e^{-\theta aF(\theta, +\infty)}\mathscr{F} (a+\tau)A(\tau) \ d\tau \ da
\end{equation}
and we have
 \begin{equation*}
   1\;=\;B\theta\int_{0}^{\infty}\int_{0}^{\infty}e^{-\theta aF(\theta, +\infty)}\mathscr{F} (a+\tau)A(\tau) \ d\tau \ da \;<\; \theta f(0) .
\end{equation*}
This means $1<\theta f(0)$, which is a contradiction.
Therefore, $F(\theta,+\infty)= 0$ for $0<\theta\leq\frac{1}{f(0)}$.\\
\noindent
\\
c) For $\frac{1}{f(0)}< \theta<1$ :  \\
Suppose  $F(\theta,+\infty)=0$.\\
From $\eqref{eq:31}$
 \begin{equation*}
   1\;\geq \;B\frac{p\theta}{1+p-\theta}\int_{0}^{\infty}\int_{0}^{\infty} e^{-\theta aF(\theta,p)}\mathscr{F} (a+\tau)A(\tau) \ d\tau \ da .
   \end{equation*}
When $p$ goes to $+\infty$,
 \begin{equation*}
1\;\geq \; \theta B\int_{0}^{\infty}\int_{0}^{\infty} e^{-\theta aF(\theta,+\infty)}\mathscr{F} (a+\tau)A(\tau) \ d\tau \ da
   \;=\; \theta f(0) \,,
   \end{equation*}
which means  $\theta \leq \frac{1}{f(0)}$ ; and this is a contradiction. Therefore, $F(\theta,+\infty) \neq 0$. \\
\noindent
\\
d) Finally, we show that $F(\theta,+\infty)$ is an  increasing function on the interval $\left(\frac{1}{f(0)},1\right)$.\\
Let us consider a function $\psi_\theta (x)$ defined for every $x \geq 0$ and $ \theta \geq 0$ by
\begin{equation*}
 \psi_\theta (x)\;=\; B\theta\int_{0}^{\infty}\int_{0}^{\infty} e^{-\theta ax}\mathscr{F} (a+\tau)A(\tau) \ d\tau \ da .
\end{equation*}
$ \psi_\theta (x)$ is well defined because $ \psi_\theta (x)\leq f(0)$.
It can be checked that $ \psi_\theta (x)$ is a strictly decreasing continuous function of $x$.
According to the Dominated Convergence Theorem (DCT)  in  $\eqref{eq:30B}$,
\begin{equation*}
\psi_\theta (F(\theta, +\infty))\;=\;1\,.
\end{equation*}
By changing the variable,  $\psi_\theta (x)$ becomes
\begin{equation*}
 \psi_\theta (x)\;=\;B\int_{0}^{\infty}\int_{0}^{\infty} e^{-bx}\mathscr{F} \left(\frac{b}{\theta}+\tau\right)A(\tau) \ d\tau \ db \,,
\end{equation*}
which  shows that $ \psi_\theta (x)$ is an increasing function of $\theta$.
Let us take $\theta_1$ and $\theta_2$ such that $\frac{1}{f(0)}<\theta_1<\theta_2<1$.
Then we have:
  \begin{equation}
\label{eq:36a}
  \psi_{\theta_2} (F(\theta_1, +\infty))\;\geq \; \psi_{\theta_1} (F(\theta_1, +\infty))\;=\;1\;=\; \psi_{\theta_2} (F(\theta_2, +\infty)) .\\     \end{equation}
 \noindent
  $\psi_\theta (x)$ is a strictly decreasing function of $x$ and from  the development \eqref{eq:36a}, we conclude that  $F(\theta_1, +\infty)\leq F(\theta_2, +\infty)$. This shows that $F(\theta, +\infty)$ is an  increasing function.\\

\subsection{Special case of natural constant per-capita mortality rate ($\mu$)}
\noindent
In this subsection, we assume that all individuals have a survival function $\mathscr{F} (a)=e^{-\mu a}$, which describes a constant per-capita mortality rate $\mu$.
\noindent
By applying the survival function, the following basic reproduction number is derived from \eqref{eq:22}:
\begin{equation}
\label{eq:23}
\begin{split}
R\;=\; f(0) \; &=\; B\int_{0}^{\infty}\int_{0}^{\infty}e^{-\mu (a+\tau)} A(\tau) \ d\tau \ da \\
      &= \;\frac{B}{\mu}\int_{0}^{\infty}e^{-z \tau}A(\tau) d\tau \, .
 \end{split}
\end{equation}
 \noindent
 This expression for the reproduction number was also found by Breda et al (2012) for a constant per-capita mortality rate $\mu$. The characteristic equation \eqref{eq:20} for endemic steady state becomes  a second degree equation. The coefficients depend on the parameters of vaccination and the reproductive number.
\begin{equation}
\label{eq:24}
\frac{\theta(1+p)}{\mu f(0)}F(\theta,p)^2 +\left(\frac{1+p+\theta}{f(0)}-\theta (1+p)\right) F(\theta,p) + \frac{\mu}{f(0)}(1-f(0)) \;=\;0 .
\end{equation}
\noindent
We define:
\begin{align*}
a & \;=\; \frac{\theta(1+p)}{\mu f(0)}  ,  \\
b & \;=\;\frac{1+p+\theta}{f(0)}-\theta (1+p) ,  \\
 c &\; =\; \frac{\mu}{f(0)}(1-f(0)) .
\end{align*}
\noindent
In an endemic steady state, $f(0)>1$, leading to $c=\frac{\mu}{f(0)}(1-f(0))<0$ and $b^2 - 4ac>0$.
\noindent
The solution of the equation \eqref{eq:24} becomes
\begin{equation}
\label{eq:26}
\begin{split}
F(\theta,p) & \;=\; \frac{-b+ \sqrt{b^2 - 4ac}}{2a} \\
   & \;=\;\frac{\mu}{2}\left\{\left(f(0)-\frac{1+p+\theta}{\theta (1+p)}\right)+ \sqrt{\left(\frac{1+p+\theta}{\theta (1+p)}-f(0)\right)^2 + 4\frac{f(0)-1}{\theta (1+p)}} \right\} .
\end{split}
\end{equation}
\noindent
The endemic force of infection is more complex  with vaccination parameters. In fact, in Breda et al (2012), with a constant per-capita mortality rate $\mu$ and no vaccination, the authors find the endemic force of infection  $F^* =\mu(f(0)-1)$.\\

\noindent
\textbf{Case 1: ineffective vaccine ($\theta =1$) }

 The endemic force of infection becomes
 \begin{equation*}
\begin{split}
F(1,p)\;=\; \lim_{\theta \to 1}F(\theta,p) \;& =\; \frac{\mu}{2}\left\{\left(f(0)-\frac{2+p}{1+p}\right)+ \sqrt{\left(\frac{2+p}{1+p}-f(0)\right)^2 + 4\,\frac{f(0)-1}{1+p}}\right\} \\
 & =\; \frac{\mu}{2}\left\{\left\{f(0)-1- \frac{1}{1+p}\right)+ \sqrt{\left(f(0)-1+ \frac{1}{1+p}\right)^2}\right\}   \\
  & =\; \mu (f(0)- 1) \\
  & =\; F^* .
    \end{split}
\end{equation*}
\noindent
In the case of 100\% ineffective vaccine, the endemic force of infection is  the same as the endemic force of infection without vaccination (Breda et al (2012)).\\

\noindent
\textbf{Case 2: 100\% effective vaccine ($\theta= 0$)}  \\
Multiplying the quadratic formula of $\eqref{eq:26}$ by $b+\sqrt{b^2-4ac}$ in numerator and denominator gives
\[
F(\theta,p) \;=\;    \frac{-b^2 +(\sqrt{b^2-4ac})^2}{2a (b+\sqrt{b^2-4ac})}
      \;=\;    \frac{-2c}{b+\sqrt{b^2-4ac}}   \,.
\]
As $\theta\rightarrow 0$, we see that $a\to 0$ and $b\to (1+p)/f(0)$, resulting in
\[    \lim_{\theta\to 0} F(\theta,p)    \;=\;   \frac{-2\mu(1-f(0))/f(0)}{2(1+p)/f(0)}   \;=\;
      \frac{\mu(f(0)-1)}{1+p}    \;=\;   \frac{F^*}{1+p}   \,.
\]
\noindent
Again, $F^* =\mu(f(0)-1)$ is the endemic force of infection without vaccination from Breda et al (2012).
Taking into account the vaccination process, we have  $F =\frac{F^*}{p+1}$, which depends on the factor
$\frac{1}{p+1}$, with vaccination rate parameter $p$. We have $\lim_{p \to \infty}\left\{\lim_{\theta \to 0}F(\theta,p)\right\} = 0$, which corresponds to the disease-free steady state.\\

\noindent
\textbf{Case 3: $p \rightarrow \infty$ and $\theta\neq0$}\\
The vaccine is not 100\% effective. By increasing the vaccination rate parameter ($p$), the expression in \eqref{eq:26} becomes
 \begin{equation*}
\begin{split}
\lim_{p \to \infty}F(\theta,p) \;& =\; \lim_{p\to\infty}\frac{\mu}{2}\left\{(f(0)-\frac{1+p+\theta}{\theta (1+p)})+ \sqrt{\left(\frac{1+p+\theta}{\theta (1+p)}-f(0)\right)^2 + 4\frac{f(0)-1}{\theta (1+p)}} \right\} \\
   & =\; \frac{\mu}{2}\left\{(f(0)-\frac{1}{\theta })+ \sqrt{\left(\frac{1}{\theta }-f(0)\right)^2 } \right\} .\\
\end{split}
\end{equation*}
\[ \therefore  \hspace{5mm}\lim_{p \to \infty}F(\theta,p)=
  \begin{cases}
  0     & \quad \text{if } \theta\leq \frac{1}{f(0)} \,, \\
 \mu(f(0)-\frac{1}{\theta })   & \quad \text{if } \frac{1}{f(0)}< \theta\leq 1\,.
  \end{cases}
\]
With a sufficiently high vaccination rate parameter ($p$), the disease-free steady state can still be approached
arbitrarily closely if the vaccine parameter ($\theta$) is below a threshold $\left(\theta\leq\frac{1}{f(0)}\right)$. \\

\noindent
\textbf{Case 4: $0<\theta<1$ and $p>0$} \\
The endemic force of infection in the expression \eqref{eq:26} was simulated as a function of vaccination rate parameter ($p$) and vaccine parameter ($\theta$). Figure \ref{fig:test2} below  provides a summary of the findings. We supposed the reproduction number ($R$) is equal to $2$.
As illustrated by the yellow area in Figure \ref{fig:test2}, the scaled endemic infection force
$\left(\frac{F^*}{\mu}\right)$ remains almost constant; whereas in the blue area,  the reduction of  the endemic force of infection is significant. Compared to the non-vaccination scenario,  the endemic force of infection is almost 0. In Figure \ref{fig:test2}, the graphs \ref{fig:1a} and \ref{fig:1b} are two views of the same function in a three-dimensional space, whereas the graph \ref{fig:1c} is a two-dimensional space.

\begin{figure}[h]
    \begin{subfigure}{.47\linewidth}
\centering
    \includegraphics[scale=0.4]{plot21}
\caption{\tiny Endemic force of infection as a function of $\theta$ and $p$}
\vspace{10pt}
\label{fig:1a}
\end{subfigure}%
\begin{subfigure}{.5\linewidth}
\centering
    \includegraphics[scale=0.4]{plot22}
\caption{\tiny Endemic force of infection as a function of $\theta$ and $p$}
  \vspace{10pt}
\label{fig:1b}
\end{subfigure}\\[1ex]
\begin{subfigure}{\linewidth}
\centering
 \vspace{-15pt}
    \includegraphics[scale=0.5]{plot23}
 \caption{\tiny Endemic force of infection in percentage (vaccination versus non-vaccination scenario)}
 \vspace{-10pt}
\label{fig:1c}
\end{subfigure}
\caption{ Impact of adaptive vaccination strategy on the endemic  force of infection ($R=f(0)=2$).}
\label{fig:test2}
 \vspace{-20pt}
\end{figure}

\section{Conclusion}
\noindent
The force  of infection  was developed by Breda et al (2012) to study the disease spread  within a closed  population  and a demographically open population. In the same context of disease spread, the susceptibles in each population are divided into two groups, non-vaccinated susceptible and vaccinated susceptible,  in order to evaluate the combined effect of vaccine effectiveness and the vaccination rate on the dynamics of the force of infection. The adaptive vaccination strategy  is used  and consists of choosing the rate of vaccination proportional to the force of infection; although simplistic,  this natural strategy makes the model more analytically tractable.
 The investigation focuses on the relation between force of infection of  the disease spreading, vaccine effectiveness, and  adaptive vaccination rate. The vaccine parameter $\theta$ ($0\leq \theta \leq 1$) captures the vaccine effectiveness, and the vaccination rate parameter ($p>0$) determines the rate of vaccination.
\noindent
 As shown by the results, the Reproduction Number ($R$) can be used to compare the model  without vaccination (Breda et al (2012)) and with vaccination. In fact in each population, $R$ is not affected by  the vaccine parameter and  the vaccination rate parameter. The findings show that the cumulative force of infection in a closed population and the endemic force of infection in a demographically open population have the same pattern of behavior as  they can be reduced significantly  by the same two factors: the effective vaccine  and the vaccination rate. In fact, for a given vaccination rate parameter ($p$), the cumulative force of infection or the endemic force of infection decreases  significantly when the vaccine is effective ($\theta$ below a threshold (dark blue color in Figure \ref{fig:test1} and Figure \ref{fig:test2})). When $p$ is sufficiently big, the threshold of effectiveness is $\frac{1}{R}$ ($\theta<\frac{1}{R}$) for  respectively closed population and  demographically open population; and the force of infection transforms an endemic steady state into to a disease-free state. The study also highlights  the fact  that only the effective vaccine is the key for  a successful reduction of the force of infection. In fact, when the vaccine is not effective ($\frac{1}{R}<\theta$), for any vaccination rate parameter  chosen, the effect on the cumulative force of infection ($y(\infty)$) or the endemic force of infection
 $\left(\frac{F^*}{\mu}\right)$ remains significantly different from 0.  The results also show that $\theta R$  serves as an effective Reproduction Number when $p$ is large.

 \section*{acknowledgements}
 \noindent
 This research was supported in part by a Discovery Grant to N. Madras from the Natural Sciences and Engineering Research Council of Canada. The authors wish to thank Mahnaz Alavinejad and Jianhong Wu for discussions of the paper of Breda et al (2012).\\

redpajama_set_name: RedPajamaArXiv


Today, you have an ample of choices for everything. This holds good even when it comes to HVAC systems!
No matter whether you are thinking to buy a new HVAC system or want to replace existing system in your commercial building, you are definitely going to encounter numerous options. Choosing the right one is quite important since you would definitely want to ensure happiness and comfort of your tenants, isn't it? Plus, you will even want the system to be practical and efficient for your building.
Here you will need to understand different types of HVAC systems commercial to help you make a wise choice. This will help you pick up just the right system for the function and size of your property. Read on to explore more about it!
This category is by far the most popular kind of HVAC since it is quite appropriate and affordable for most of the smaller commercial buildings. They are perfect for the buildings having small server rooms, offices, or small-scale restaurants as they provide cooling and heating control in every room.
It is quite cheaper as compared to installing central system.
Being self contained systems, even if one unit tends to break down, the others aren't affected. They can still be functional.
With multi split system, you can easily connect around 9 indoor units with 1 outdoor unit. These systems basically work with the help of inverter technology. This technology facilitates compressor to operate at various speeds.
Each time compressor is turned on or off, huge amount of the energy is being consumed in the forced-air HVAC. However, multi-split system has sensors that detect changes in the temperature and then adjusting whenever required. As a result, this system cuts down on energy.
You can use ceil mounted as well as wall mounted air conditioning units.
You can preserve look of the building and outdoor space with much fewer outdoor units.
Variable Refrigerant Flow (VRF) or Variable Refrigerant Volume (VRV), both addresses same kind of HVAC system. If your commercial space is large or medium like large office space or mixed-use building, then going with VRV or VRF system can be your best bet.
Knowing all your available options is always the best idea. Moreover, determine the required features that can suit your tenants as well as your wallet!
Are You Aware of Different Types of Foundation Repair?
Options for repairing of foundation varies greatly based upon the geographical conditions of the area. Till 1970, people usually repaired the foundation by using only concrete, however nowadays many other types of foundation repairs are available.
Following are few different approaches for repairing the foundation.
When this method was introduced it really revolutionized the repair process for foundation. It is possible to do the repair work much faster as compared to traditional concrete pier method of repairing. Nowadays a lot of innovations have been done on this method and it has become a data driven process.
Whether it is internal slab repair or external foundation repair, helical pier can work well for both the areas. Also, if it is any new construction or for any foundation repair, it is possible to employ this method. Porch column areas or structures which are having vinyl siding that are light-loaded areas are the right place for undertaking this method. This method is quite versatile however much underused in the market.
Most of the foundation companies use this pour-in-place technique by using concrete piers during pre-construction phase of any building structures. The same methods are also used during repair of foundations too. Before the hydraulic driven steel pier method was in the market, people used to adopt this method only.
This is a permanent way of repairing the foundation however there are few disadvantages. The difficulties in drilling in any residential area and also the cost involved for this process limits this method. There are lots of digging need to be done and lots of dirt also has to be removed and after that the clean-up process also needs quite a lot of time and effort.
This is another new method which has been recently developed and has become quite popular in the market because of its speed of repair and the price that need to be spend. Slab repair has become quite simple by using such high-density polyurethane foam.
Here the repair technician will inject the polyurethane foam inside the checkerboard grid which can be about six feet on the center of the area which is affected. However, while implementing this process one has to be careful about the HVAC ducts that may be running under the floor and foam may often clog them. Also, before undertaking the job, plumbing testing is necessary to check for any leakages in the drain or supply lines.
Its is not very simple and straightforward to install split air conditioner in any room. You will get right kind of cooling effect only if it is properly installed by the service provider. However, if the work is not done in an accurate manner then the desired cooling effect in the room will be missing. There can also be frequent maintenance related issues too.
You may also visit the website gnrcorbus.com to know about various options available to you for air conditioning. Following are few things that you need to consider while installing any split air conditioner in the house.
The split air conditioner has certain amount of weight, that the wall needs to hold and hence construction of the wall must be strong enough for that.
There is a requirement of minimum 15 cm of open space from all sides of the air conditioner so that there is proper flow of air.
In order to get adequate cooling effect, it is essential that the air conditioner must be installed at a height of 7 to 8 feet above the ground level.
When you are fixing the bracket on the wall then you must make sure that the bracket must be installed by providing certain amount of tilt so that the air conditioner unit is also fitted at certain angular fashion and there is unrestricted flow of condensed water while draining out from the pipe.
If you want to achieve proper cooling then it is necessary that the outdoor unit must be installed in such location where direct sunlight or rainwater may not fall. Also, it is important that the outdoor unit must be located in any open space so that heat can be properly dissipated from the condenser unit.
It is necessary that outdoor unit must be fitted in any rigid and flat surface. This unit consists of condenser, compressor and fan motors and hence the surface where it is mounted need to be flat, otherwise it may vibrate too much and the copper pipe may get broken and also disturb you.
In order to reduce the loss of cooling the distance between indoor and outdoor unit must be minimum and never exceed more than 15 meters.


Julie Davies recently id a past event at UCLA called "Strategies For Integrating Faith and Mental Wellness" that I spoke at 2 workshops on Tuesday June 6 from 7:30-9pm.
Christy Pierce was recently interviewed by TBN. Check out the entire interview here or if you prefer audio, you can listen to the audio version.


text:
Otha Foster III (born September 27, 1988) is an American professional gridiron football defensive back currently playing for the Saskatchewan Roughriders of the Canadian Football League. He played college football at West Alabama. He has been a member of the Kansas City Chiefs and Baltimore Ravens of the NFL, and the Toronto Argonauts, Edmonton Eskimos and BC Lions of the CFL.

College career
Foster played two years at Pearl River Community College during 2009 and 2010. He then transferred to the University of West Alabama where he completed his college career in 2011 and 2012, where his career record included 96 tackles, 16 pass deflections and 10 interceptions.

Professional career

Kansas City Chiefs
In May 2013, Foster signed with the Kansas City Chiefs of the NFL, as an undrafted free agent He played three preseason games with the Chiefs before being cut from the team.

Toronto Argonauts
He then moved to the Canadian Football League, signing with the Toronto Argonauts from whom he was acquired by the Edmonton Eskimos in 2014. He was traded to Edmonton from Toronto on May 13, 2014 along with Toronto's first and second round picks (6th and 15th overall) in the 2014 CFL Canadian Draft, import offensive lineman Tony Washington and a negotiation list player, in exchange for Edmonton's first and third round picks (3rd and 21st overall) in the 2014 CFL Canadian Draft and a negotiation list player.

Edmonton Eskimos
After being traded to the Eskimos Foster became an immediate starter, and played in 16 games in the 2014 season. Foster recorded 46 defensive tackles, five special teams tackles, five pass knockdowns, one interception, one sack, one fumble recovery and one forced fumble. In the Labour Day Classic, Foster recovered a blocked punt by Willie Jefferson and scored a 57-yard touchdown. In the West Semi-final against Saskatchewan, Foster recorded one defensive tackle. In the West Final against Calgary, he recorded two defensive tackles. Foster continued his strong play in the 2015 season, contributing 57 tackles, 2 special teams tackles, 2 sacks, 1 interception, 1 forced fumble and 1 defensive touchdown.

Saskatchewan Roughriders
On February 10, 2016, Foster signed as a free agent with the Saskatchewan Roughriders of the CFL. Foster had his best statistical season in 2016 playing in all 18 regular season games and racking up 71 tackles and 3 quarterback sacks. Set to become a free-agent in February 2017, Foster had a workout with the New Orleans Saints In December 2016. On January 20, 2017 Foster had a workout with the Baltimore Ravens. On January 27, 2017, a couple weeks before becoming a free agent, Foster was released by the Riders so he could pursue NFL opportunities.

Baltimore Ravens
On January 31, 2017, Foster signed a reserve/future contract with the Baltimore Ravens. He was waived on September 1, 2017 during final roster cutdowns.

Saskatchewan Roughriders
He signed with Saskatchewan in September 2017.

BC Lions
Foster signed with the BC Lions on February 14, 2018.

Personal life
Foster joined the United States Marine Corps in 2007 after he was redshirted at Southern University, attending basic training at Marine Corps Recruit Depot Parris Island and becoming a communications expert. He served in the Marines while playing football for two years at Pearl River Community College.

He is the father of two children, Olavia K. Foster, born on April 16, 2010. He is also the father of Otha Foster IV, born on Foster's Mother is Valerie Lee Foster Aikens. Foster's father, Otha Foster Jr., died January 9, 2010 after being a disabled dialysis patient for 19 years. Foster III was the main caregiver for his father since the age of eight. Even though Foster Jr. was on dialysis, he still managed to coach Foster III's little league football team. He lived to see him play JUCO, but he died at the age of 43, during Christmas break, before Foster III started at West Alabama.

References

External links
 BC Lions bio

Living people
1988 births
American football defensive backs
Canadian football defensive backs
American players of Canadian football
Pearl River Wildcats football players
People from Washington Parish, Louisiana
Players of American football from Louisiana
West Alabama Tigers football players
Toronto Argonauts players
Edmonton Elks players
Saskatchewan Roughriders players
Baltimore Ravens players
BC Lions players

redpajama_set_name: RedPajamaWikipedia


The FO Resource & FAQ Page is intended for Facilities Operations personnel reference only.
Here you can find answers to frequently asked questions and other FO resources and/or links.
Here you can find information regarding access to ARCHIBUS, as well as other basic FAQs related to ARCHIBUS.
Please complete the webform for Access to ARCHIBUS and S Drive.
What levels will be granted in ARCHIBUS based on my position level (i.e. Manager, Supervisor, Admin, Tech, etc)?
Disposing and transferring multiple items from your department.
Hiring a new, existing, or transfer employee into your work group.
Requests for Banner Finance/49er Mart Access.
Sick leave, vacation, or bonus leave that may be donated (The Voluntary Shared Leave Program).
The granting of a waiver from the Competitive Solicitation Process.
Notifying the Purchasing Department that goods need to be returned to a vendor.
Using the correct incident reporting procedure.
Here you will find Facilities Operations Policies and Procedures.
Here you can find information regarding access to the S Drive.
How do I grant/remove/update an employees' access to the S Drive?
What folders will I be able to see on the S Drive based on my position (i.e. Supervisor, Admin, Tech, etc)?
Here you can find information regarding access and facts to SDI Exchange.
How do I grant/remove/update an employees' access to SDI Exchange?
I have an After-Hours Emergency Purchase Request. What is the process for this?
I have a part(s) status incorrectly listed on a work order. How can I fix this? Please contact Pamela Duff at 7-0552.
I will be on vacation/away from my office and need my orders to be approved by someone else. What do I do? Please contact Pamela Duff at 7-0552.


text:
MBNA Thames Clippers
Home	Contact Us	About MBNA Thames Clippers
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From palaces and cathedrals to ships and prisons, historic buildings and tours offer a great way to learn about the city's past.
St. Paul's Cathedral
Bankside Pier
Enter St. Paul's and see the cathedral's awe-inspiring interior. You can use a touch-screen multimedia guide or join a guided tour to discover the tombs and memorials of some of the nation's greatest heroes, such as Admiral Lord Nelson and the Duke of Wellington. There's also an award-winning 270° immersive film experience. Check out the acoustic quirks of the Whispering Gallery and continue the climb to the Golden Gallery for panoramic views across London.
Walking distance: 15 minutes
Golden Hinde
The Golden Hinde is a historically accurate replica of the ship in which Sir Francis Drake completed the second ever circumnavigation of the world between 1577-1580. The ship is open to the public for guided tours with costumed actors and it is even possible to join Drake and his crew on an overnight voyage of adventure.
Shakespeare's Globe
Visit Bankside, the Soho of Elizabethan London and experience today's working theatre. Shakespeare's Globe is a faithful reconstruction of the open-air playhouse designed in 1599 where many of Shakespeare's plays were performed. Shakespeare's Globe is open year-round, with an exhibition, theatre tour, educational workshops and an outdoor and indoor theatre. It also has a shop, bar and brasserie.
Watch this video to find out more about the Exhibition and Tour.
Greenwich Royal Tours
Greenwich Pier
Greenwich was designated a World Heritage Site in 1997. Major venues include: Cutty Sark, Greenwich Royal Observatory, National Maritime Museum, Queen's House and the Old Royal Naval College. Greenwich Royal Tours do guided walking tours with all admissions included, with full-day, half-day and evening pub tours available.
Southwark Cathedral
London Bridge City Pier
Southwark Cathedral is London's oldest Gothic church. It has connections with William Shakespeare, US university benefactor John Harvard and the first English poet, John Gower. Drop by on Monday lunchtimes for an organ recital or Tuesday afternoons at 3:30pm for a short music recital.
Rose Theatre Exhibition
The Rose, built in 1587, was the first Elizabethan theatre on Bankside and was where Shakespeare learnt his craft. Shakespeare's Globe exhibition offers tours of the nearby Rose Theatre site to pre-booked groups. The Rose Theatre also hosts a series of theatrical performances.
Museum of London Docklands
Canary Wharf Pier
From Roman settlement to modern regeneration, the Museum of London Docklands reveals the long history of London as a port through stories of trade, migration and commerce. With archaeological finds, fascinating tours, films and events, this is one of the capital's real hidden gems.
Following an ambitious six-year conservation project Cutty Sark reopened in April 2012. The ship has been raised over three metres allowing visitors the unique and jaw-dropping experience of walking underneath a nineteenth century sailing ship, the last surviving tea clipper, and the fastest and greatest of her time.
The Old Royal Naval College (ORNC) is Sir Christopher Wren's twin-domed riverside masterpiece and one of London's most famous landmarks.
The iconic buildings stand on the site of Greenwich Palace, Henry VIII's favourite royal residence, and include the Discover Greenwich Visitor Centre, Painted Hall and Chapel. You can find out more about the ORNC's intriguing past by joining one of their daily guided tours.
Churchill War Rooms
Westminster Pier
Visit Churchill War Rooms, part of Imperial War Museums, to discover the original Cabinet War Rooms, the wartime bunker which sheltered Churchill and his staff during the Blitz. Explore the historic rooms to experience the secret history that lives on underground. Look through the lens of history into the Map Room where the books and charts have remained exactly where they were left when the door was locked in 1945. Discover the stories of those who worked underground as London was being bombed above them, and explore the life and legacy of Winston Churchill in the interactive Churchill Museum.
Creepy London
London Attractions For Kids
London Markets and Shopping
London Views
Places To Go In London
Sightseeing and Leisure
What's On and Discounts
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var axel = Math.random() + ""; var a = axel * 10000000000000; document.write(' ');

redpajama_set_name: RedPajamaCommonCrawl


text:
Immersives
Top Escape Rooms of 2018
Our Story So Far:
Discovering the world of escape rooms in 2016 was groundbreaking for us. Since then, we've launched our review website, joined some awesome online communities, and traveled across the country (and out of the country) to take part in some unforgettable experiences.
We are still growing and still learning every day but our current stats have us standing at almost 100 rooms completed with a 95% escape rate.
Some cities outside of our home state of New Jersey that we were able to travel to for rooms this year include:
As this is our first year issuing rankings, the following list will include rooms that we completed in the time between our conception in 2017 and current-day, 2018.
Another noteworthy parameter is that as of this publishing date, all of the rooms on this list are still open. Rooms that were closed prior to December 2018 were removed from consideration.
We've done some incredible rooms over this almost two-year period. Rooms that did not make this list are surely not considered unnoteworthy. However we wanted to take the time to shout-out some of the ones that we enjoyed the most.
Top Rooms of 2018
Ranked in alphabetical order
Battleship – Unreal Escapes Staten Island, NY
This long-awaited room really showcased the amount of research, detail, and effort that was put into it. The incredible set design made for believable gameplay and an arching narrative lead us through a great story where puzzles were driven by our interactions with the ship.
Cursed – Komnata Quest – Manhattan, NY
The most frightening room that we've done on the east coast. Komnata crafted a game that had the blood drain from our faces right from the start. Surprises met us around every corner as we explored the abandoned home where seven sisters once lived.
Elevator Shaft – The BASEMENT – Los Angeles, CA
THE BASEMENT took a space the size of, well, an elevator shaft and filled it with so much wow-factor. Adding in impressive production elements including smell, sound, and even some water features, this experience really transferred us out of our world and into the story.
Escape the Pharaoh's Tomb – Locked Up Escape Games – Buffalo, NY
For all of the tombs that we've escaped, this one felt the most immersive and daunting. Our group cannot wait to make it back to Buffalo for more Locked Up Escapes!
High Speed – Exit Escapes -Manhattan, NY
Another room with an incredibly accurate set, High Speed NYC had us journey through an authentic New York City subway system. Puzzles aligned with actions we would take on an actual subway car.
Lab Rat – Hatch Escapes – Los Angeles, CA
With roles reversed, we found ourselves as part of an experiment run by Doctor Ratkenstein, tasked to complete a series of puzzles. Between strategic and colorful lighting combined with musical queues, this room crafted a level of theatrics that had us feeling as though we were working our way through a larger than life world with childlike wonder.
Sanatorium – I Survived the Room – Long Island City, NY
One of the best actor-driven rooms that we've done, Sanatorium also masterfully executed a game that split us up but still forced us to collaborate in order to find success. We'll never forget the dramatic ending.
Stash House – Los Angeles, CA
Stash House had us feeling like we were in The Wire. With an ever-expanding set that incorporated thematic and unique puzzles, our time within Ray's trap house built up to a watery end.
The Courtyard – The BASEMENT – Los Angeles, CA
THE BASEMENT had topped even themselves by delivering the most impeccable set we'd seen through their awe-inspiring execution of Edward Tandy's courtyard. The sights and sounds were elevated through the theatrical use of an actor who assisted as a built-in game master, storyteller, and even functioned as the main focal point for an incredibly memorable mid-game puzzle.
Zoe – Escapade Games – Fullerton, CA
Our journey through Zoe took the crown to be the more terrifying haunt we'd experienced to date. We shuddered through her home working through puzzles in varying levels of darkness. Expect the unexpected.
Congratulations to each winner! We are excited to see what 2019 has in store!
Check out some of our Recommendations!
New Jersey Escape Room Recommendations
Benji's Favorite Five Escape Rooms
New York City Escape Room Recommendations


Colonel Randall Larsen, USAF (Ret), is the Director of The Institute for Homeland Security and the National Security Advisor to the Center for Biosecurity of the University of Pittsburgh Medical Center.
Colonel Larsen began his studies of homeland security in 1994 while on a one-year fellowship at the University of Pittsburgh Graduate School of Public and International Affairs. In 1999, while serving as the Chairman of the Department of Military Strategy and Operations at the National War College, he created the nation's first graduate course in homeland security.
Larsen was one of the first witnesses to testify before the 9-11 Commission, and since 9-11 he has served as an expert witness to the Senate Armed Services, Senate Judiciary, Senate Homeland Security, House Government Reform, House Homeland Security, and House Budget Committees.
In 2009, he served as the Executive Director of the Congressional Commission on the Prevention of Weapons of Mass Destruction Proliferation and Terrorism. In 2010, former US Senators Bob Graham (D-FL) and Jim Talent (R-MO) founded the Bi-Partisan WMD Terrorism Research Center to continue the work of the WMD Commission. Larsen served as the CEO of the WMD Center until December 2012.
Larsen was the co-host of the weekly, one-hour public radio show, Homeland Security: Inside and Out. He is also a frequent guest on radio and television including: CBS, NBC, ABC, CNN, BBC, NPR, and The Jim Bohannon show.
Colonel Larsen served in both the Army and Air Force for a combined total of 32 years of military service. He began his flying career as a 19-year old Cobra pilot in the 101st Airborne Division, where he flew 400 combat missions in Vietnam. He also served as military attaché at the U.S. Embassy in Bangkok, the chief of legislative liaison at the U.S. Transportation Command, and the commander of America's fleet of VIP aircraft at Andrews AFB MD. His decorations include the Legion of Merit, Distinguished Flying Cross, Bronze Star, 17 awards of the Air Medal (3 with "V" Device for Valor), and the South Vietnamese Cross of Gallantry.


Originally from the town of Dombivli near Mumbai, Tejashree completed her education from the Chandrakant Patkar Vidyalaya.
Hya Gojirvanya Gharat was one of her first serials. She later played lead role in ETV Marathi's Lek Ladki where she played the role for more than 800 episodes. However, she truly found big fame with the role of Janhvi in superhit serial on Zee Honar Sun Me Hya Gharchi. Her pairing with Shashank Ketkar, who played the role of Shri, made the Shri – Janhvi jodi a hit among the audience.
Though mostly popular for her TV roles, Tejashree was also seen in the movie Dr Prakash Amte as the young Manda Amte. She also stars in a natak along with Prashant Damle called Karti Kaljat Ghusli.


Hi all, and thanks again for reading. There's been quite a bit happening this past month, so read on and I might even give you a chance to win a digital copy of one of my books!
The biggest announcement this month was declaring my involvement in Crystal Lake Publishing's forthcoming non-fiction anthology, It's Alive: Bringing Your Nightmares to Life.
Clive Barker, Chuck Palahniuk, Michael Bailey, F. Paul Wilson, Kevin J. Anderson, Christopher Golden, Rachel Autumn Deering, Yvonne Navarro, Greg Chapman, Tom Monteleone, Lisa Morton, John Palisano, Joe R. Lansdale and Kasey Lansdale, Todd Keisling, Richard Thomas, Brian Kirk, Elizabeth Massie, Lisa Mannetti, Jess Landry, David J. Schow, Tonya Hurley, Linda D. Addison, Cody Goodfellow, Stephanie M. Wytovich, Tim Waggoner, David Wellington, Elizabeth Massie, Bev Vincent, James Chambers, Stephen Graham Jones, Vince A. Liaguno, and Jonathan Maberry and Mary Sangiovanni! With an introduction by Richard Chizmar. What a line-up right?
My essay centres on how I penned my debut novel, Hollow House and how I'd agonised for years in taking the leap to write it.
The anthology will be released on December 14 and it's already picking up some rave reviews, so if you're a writer, you should definitely have this on your shelf.
Back in 2014, I had my debut short story collection published – Vaudeville and Other Nightmares and I've decided to resurrect it with a new look, and new(ish) stories.
To be published by Specul8 Publishing here in my hometown of Rockhampton, the new edition will feature 30 stories, including the novella Vaudeville, with each story illustrated by yours truly.
The book is expected to be released around April 2019, so please watch this space for more details.
But to give you a taste of what to expect I've included an image of one of the pieces of internal art opposite.
And finally, the re-release of my very first published novella, Torment, happened this month. The fine folks at Omnium Gatherum Books have done a fantastic job giving this story a new lease on life.
Available now in digital and paperback formats, the novella is a creepy haunting/exorcism tale set in Scotland.
The Horror Fiction Review has said that "Torment features some genuinely scary moments and keeps you guessing…" so I hope you'll check it out!
Horror is actually good for you!
I've always maintained that watching (and reading horror) can be beneficial to your health and this article about a university study into the health benefits of watching horror flicks supports that view.
Research has found that watching horror movies releases neurotransmitters, which not only increase brain activity, but expand our ability to think! It also apparently burns as much calories as taking a 30-minute walk!
There's been a number of issues with Amazon and Australia of late and (until recently) it wasn't exactly good news if you're an author, publisher, (or a customer).
So last year the Australian Government decided to impose GST (Goods and Services Tax) on Amazon.com for goods purchased from their site and shipped to Australia. In response, Amazon restricted access to the Amazon.com site and set up an Australian-based site Amazon.com.au. The issue meant some items were not available and publishers were no longer able to ship direct contributor copies to their Australian-based authors without incurring an additional cost.
The good news is that Amazon.com appear to have since waived that decision to exclude Australians from the .com site. But I'm still not sure whether publishers can send books direct to Australia. Hopefully some clarification will be available on that soon.
So I work at a university and their engineering labs have a bunch of 3D printers. Because they're always looking for a challenge, I asked if they could print a few horror-themed items for me that I could paint. Check out the results below!
Want to win a digital copy of my Bram Stoker Award-nominated novel, Hollow House? Just subscribe to my blog to go into the draw.
Feel free to share and thanks for subscribing!
This entry was posted in Uncategorized and tagged 3D printing, amazon, crystal lake publishing, fiction, horror, omnium gatherum books, specul8 publishing. Bookmark the permalink.


text:
https://warriorswire.usatoday.com/2022/02/22/report-warriors-could-attempt-to-sign-guard-jordan-poole-to-extension-during-offseason/
Report: Warriors could attempt to sign guard Jordan Poole to extension during offseason
Daniel Dunn-USA TODAY Sports
While there are several key factors in the Golden State Warriors' surge to the second-best record in the NBA during the 2021-22 season, Jordan Poole's rise has played a significant role.
After a strong finish to his sophomore campaign, Poole entered the starting lineup with Klay Thompson pushing through the final stages of his injury rehab. In his third season since being drafted in the first round of the 2019 draft, the former Michigan player is averaging 16.4 points on 43.8% shooting from the field with 3.5 assists and 3.1 rebounds in 28.8 minutes per contest.
Along with averaging career-highs in points, field goal shooting percentage, rebounds, assists and minutes, Poole is leading the league from the free-throw stripe, shooting 92.2% from the line.
Following Poole's breakout, the Warriors could be planning to lock up the 22-year-old guard during the offseason. According to Connor Letourneau of The San Francisco Chronicle, the Warriors will attempt to sign Poole to a multiyear extension during the summer.
Via The San Francisco Chronicle:
The Warriors also will try to sign Poole to a multiyear extension this summer, which would allow them to avoid having to deal with him in restricted free agency in 2023. Keith Smith, a salary-cap expert for Spotrac, estimates that deal to be in the four-year, $80 million range.
While the Warriors will have to get creative when it comes to the salary cap with Steph Curry, Klay Thompson and Draymond Green all under contract, finding a way to extend Poole before he hits the open market could be the route to locking up a key piece of their young core.
As the Warriors build toward a potential deep run through the postseason, Steve Kerr will continue to lean on Poole in a vital role off the bench down the stretch.
NBA Twitter reacts to Steph Curry's record-breaking MVP performance at 2022 All-Star Game


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Q: How to not refresh a page when a button is pressed I have been working on a like/unlike function on my django project in which a user can like like a post and if so the user can dislike it like in instagram. Now to finish with this part of the project I need to make the not refresh the page when a like or unlike button is pressed but the problem is that I am very very new to ajax. How can I acomplish this?
models.py
class Post(models.Model):
    text = models.CharField(max_length=200)
    user = models.ForeignKey(User, related_name='imageuser', on_delete=models.CASCADE, default='username')
    liked = models.ManyToManyField(User, default=None, blank=True, related_name='liked')
    updated = models.DateTimeField(auto_now=True)
    created =models.DateTimeField(auto_now_add=True)

    def __str__(self):
        return str(self.text)

    def get_absolute_url(self):
        return reverse('comments', args=[self.pk])

LIKE_CHOICES = (
    ('Like', 'Like'),
    ('Unlike', 'Unlike'),
)

class Like(models.Model):
    author = models.ForeignKey(User, on_delete=models.CASCADE)
    post = models.ForeignKey(Post, on_delete=models.CASCADE)
    value = models.CharField(choices=LIKE_CHOICES, default='Like', max_length=10)

    def __str__(self):
        return str(self.post)

views.py
def like_post(request):
    user = request.user
    if request.method == 'POST':
        post_id = request.POST.get('post_id')
        post_obj = Post.objects.get(id=post_id)

        if user in post_obj.liked.all():
            post_obj.liked.remove(user)
        else:
            post_obj.liked.add(user)

        like, created = Like.objects.get_or_create(author=user, post_id=post_id)

        if not created:
            if like.value == 'Like':
                like.value == 'Unlike'
            else:
                like.value = 'Like'

        like.save()
    return redirect('home')

def home(request):
    contents = Post.objects.all()

    context = {
        "contents": contents,
    }
    print("nice2")
    return render(request, 'home.html', context)

urls.py
urlpatterns = [
    path('', views.home, name='home'),
    path('like/', views.like_post, name="like-post"),
]

home.html
<form action="{% url 'like-post' %}" method="POST">
  {% csrf_token %}
  <input type='hidden' name="post_id" value="{{ content.id }}">
  {% if user not in content.liked.all %}
    <button type="submit">Like</button>
  {% else %}
    <button type="submit">Unlike</button>
  {% endif %}
</form>
<strong>{{ content.liked.all.count }}</strong>

A: To avoid updating the page, you can set the button on your html as:
 <button type='button'>submit</button>

Instead of using submit, you'll have to add an event listener to the button or a function associated with the onclick so you can call the ajax to handle this request on the background.
You can also achieve that by providing a null action for you form, by using:
<form action="javascript:void(0);" method="POST">

Take a look at this tutorial:
https://simpleisbetterthancomplex.com/tutorial/2016/08/29/how-to-work-with-ajax-request-with-django.html

A: This is related to the default html behavior of a button with type=submit inside a form tag. By default, if you click in the button with type submit, it tries to send your data and reload your page. You can just change the type of the button and let your javascript code deal with the button behavior. On each of your buttons, add:
<button id="like-button" type="button">Like</button>

For the JavaScript code, just add to your  tag or separate js file:
window.addEventListener('DOMContentLoaded', (event) => {
    let likeButton = document.querySelector(#like-button);
    likeButton.addEventListener('click', () => {
          your code when like button is clicked
    }
});

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