\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada.
\newline {\em Electronic Journal of Differential Equations},
Conference 12, 2005, pp. 29--37.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{29}
\begin{document}
\title[\hfilneg EJDE/Conf/12 \hfil Age of infection]
{Age of infection in epidemiology models}
\author[F. Brauer \hfil EJDE/Conf/12 \hfilneg]
{Fred Brauer}
\address{Department of Mathematics \\
University of British Columbia \\
Vancouver, BC, V6T 1Z2, Canada}
\email{brauer@math.ubc.ca}
\date{}
\thanks{Published April 20, 2005.}
\thanks{This work was supported by MITACS and by an NSERC grant}
\subjclass[2000]{92D30}
\keywords{Epidemic; age of infection; endemic equilibria}
\begin{abstract}
Disease transmission models in which infectivity depends on the time
since infection are of importance in studying such diseases as HIV/AIDS.
They also provide a means of unifying models with exposed stages or
temporary immunity. We formulate a general age of infection model and
carry out a partial analysis. There are open questions in the analysis
of the characteristic equation at an endemic equilibrium.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\section{Introduction}
The 1927 epidemic model of Kermack and McKendrick \cite{KMcK27} is considerably more general
than what is usually called the Kermack-McKendrick epidemic model. The general model described by Kermack and McKendrick included a dependence of infectivity on the time since becoming infected (age of infection). The 1932 and 1933 models of Kermack and McKendrick
\cite{KMcK32, KMcK33},
which incorporated births and deaths, did not include this dependence. Curiously,
while age of infection models have not played a role in studies of epidemics, they
are very important in studies of HIV/AIDS. Since HIV/AIDS acts on a very long time
scale it is essential to include demographic effects (recruitment into
and departure from a population of sexually active individuals). Also, the
infectivity of HIV-positive people is high for a relatively short time after
becoming infected, then very low for a long period, possibly several years, and
then high shortly before developing into full-blown AIDS. Thus, the age of
infection for models described by Kermack and McKendrick for epidemics but not for
endemic situations, have become important in endemic situations.
We will describe a general age of infection model which includes demographic
effects and carry out a partial analysis. There are many unsolved problems in
the analysis, centered on the analysis of the characteristic equation at an
endemic equilibrium.
\section{The Basic $SI^*R$ Model}
We let $S(t)$ denote the number of suceptibles at time $t$ and
$R(t)$ the number of members recovered with immunity, as is
standard in compartmental epidemiological models. However, instead
of using $I(t)$ to denote the number of infective members at time
$t$ we let $I^*(t)$ denote the number of infected (but not
necessarily infective) members and let $\phi (t)$ be the total
infectivity at time $t$.
We make the following assumptions:
\begin{enumerate}
\item The population has a birth rate $\Lambda (N)$, and a natural death
rate $\mu$ giving a carrying capacity $K$ such that $\Lambda(K)= \mu K,
\Lambda '(K)<\mu $.
\item An average infected member makes $C(N)$ contacts in unit time of which $S/N$
are with susceptible. We define $\beta(N)=C(N)/N$ and it is
reasonable to assume that $\beta ' (N) \leq 0, C'(N) \geq 0$.
\item $B(\tau)$ is the fraction of infected individuals remaining infective if alive when
infection age is $\tau$ and $B_\mu (\tau) = e^{-\mu \tau} B(\tau)$ is the
fraction of infected ones remaining alive and infected when infection age is
$\tau$. Let $\hat{B_\mu}(0) = \int_0^\infty B_\mu (\tau) d\tau$
\item A fraction $f$ of infected members recovers with immunity and a
fraction $(1-f)$ dies of disease.
\item $\pi (\tau)$ with $0 \leq \pi (\tau) \leq 1$ is the infectivity at
infection age $\tau$; let $A(\tau) = \pi(\tau) B(\tau),
A_\mu(\tau) = \pi(\tau) B_\mu (\tau),
\hat{A_\mu}(0) = \int_0^\infty A_\mu (\tau) d\tau $.
\end{enumerate}
We let $i_0(t)$ be the number of new infected individuals at time
$t$, $i(t, \tau)$ be the number of infected individuals at time
$t$ with infection age $\tau$. Then
\begin{gather*}
i(t, \tau) = i_0(t - \tau) B_\mu (\tau), \quad 0 \leq \tau \leq t \\
i_0(t) = S \beta (N) \phi (t)
\end{gather*}
and
\begin{align*}
S' &= \Lambda (N) - \mu S - \beta (N) S \phi \\
I^*(t) &= \int_0^\infty i(t, \tau)d\tau \\
&= \int_0^\infty i_0 (t- \tau) B_\mu (\tau)d\tau \\
&= \int_0^\infty \beta (N(t - \tau)) S(t - \tau)\phi
(t - \tau) B_\mu(\tau) d\tau \\
\phi(t) &= \int_0^\infty i_0 (t- \tau) A_\mu (\tau)d\tau \\
&= \int_0^\infty \beta (N(t - \tau)) S(t - \tau)\phi
(t - \tau) A_\mu(\tau) d\tau
\end{align*}
Differentiation of the equation for $I^*$ shows that the rate of recovery plus the
rate of disease death is
\[
-\int_0^\infty \beta (N(t - \tau)) S(t - \tau)\phi (t - \tau) e^{-\mu\tau}B'(\tau) d\tau
\]
Thus the $SI^*R$ model is
\begin{equation} \label{age}
\begin{aligned}
S' &= \Lambda (N) - \mu S - \beta (N) S \phi \\
\phi(t) &= \int_0^\infty \beta (N(t - \tau)) S(t - \tau)\phi (t - \tau)
A_\mu(\tau) d\tau\\
N'(t) &= \Lambda (N) - \mu N
+ (1 - f)\int_0^\infty \beta (N(t - \tau)) S(t - \tau)\phi (t - \tau)
e^{-\mu \tau}B'(\tau) d\tau
\end{aligned}
\end{equation}
Since $I^*$ is determined when $S, \phi, N$ are known we have dropped
the equation for $I^*$ from the model, but it will be convenient to recall
\[
I^*(t) = \int_0^\infty \beta (N(t - \tau)) S(t - \tau)\phi (t - \tau) B_\mu(\tau) d\tau
\]
If $f = 1$ then $N(t)$ approaches the limit $K$, the model is asymptotically
autonomous and its dimension may be reduced to $2$, replacing $N$ by the constant $K$.
We note, for future use, that
\[
\hat{B_\mu}(0) = \int_0^\infty e^{-\mu\tau}B(\tau)d\tau \leq
\int_0^\infty e^{-\mu\tau}d\tau = 1/\mu,
\]
so that
$0 \leq 1 - \mu \hat{B_\mu}(0)\leq 1$.
We define $M = (1 - f)(1 - \mu \hat{B_\mu}(0)$, and $0 \leq M \leq 1$. We note,
however, that if $f = 1$ then $M = 0$.
We also have, using integration by parts,
\[
- \int_0^\infty e^{-\mu\tau}B'(\tau)d\tau = 1 - \mu \hat{B_\mu}(0) \geq 0
\]
If a single infective is introduced into a wholly susceptible population,
making $K\beta(K)$ contacts in unit time, the fraction still infective at
infection age $\tau$ is $B_\mu (\tau)$ and the infectivity at infection
age $\tau$ is $A_\mu (\tau)$. Thus $R_0$, the total number of secondary
infections caused, is
\[
\int_0^\infty K\beta(K) A_\mu (\tau) d\tau = K\beta(K)\hat{A_\mu}(0)\,.
\]
\section{Exposed periods}
One common example of an infection
age model is a model with an exposed period, during which individuals have been
infected but are not yet infective. Thus we may think of infected susceptible
individuals going into an exposed class ($E$), proceeding from the exposed class
to the infective class ($I$) at rate $\kappa E$ and out of the infective
class at rate $\alpha I$. Exposed members have infectivity $0$ and
infective members have infectivity $1$. Thus $I^* = E + I$ and $\phi = I$.
We let $u(\tau)$ be the fraction of infected members with infection age $\tau$
who are not yet infective if alive and $v(\tau)$ the fraction of
infected members who are infective if alive. Then the fraction
becoming infective at
infection age $\tau$ if alive is $\kappa u(\tau)$, and we have
\begin{equation} \label{exp}
\begin{gathered}
u'(\tau) = - \kappa u(\tau), \quad u(0) = 1 \\
v'(\tau) = \kappa u(\tau) - \alpha v(\tau) \quad v(0) = 0\,.
\end{gathered}
\end{equation}
The solution of the first of the equations of (\ref{exp}) is
$u(\tau) = e^{-\kappa\tau}$
and substitution of this into the second equation gives
\[
v'(\tau) = \kappa e^{- \kappa\tau} - \alpha v(\tau)
\]
When we multiply this equation by the integrating factor $e^{\alpha \tau}$
and integrate, we obtain the solution
\[
v(\tau) = \frac{\kappa}{\kappa - \alpha}[e^{-\alpha \tau} - e^{- \kappa \tau}]
\]
and this is the term $A_\mu (\tau)$ in the general model. The term $B(\tau)$ is
$u(\tau) + v(\tau)$.
Thus we have
\begin{gather*}
A(\tau) = \frac{\kappa}{\kappa - \alpha} [e^{- \alpha \tau} - e^{- \kappa \tau}] \\
B(\tau) = \frac{\kappa}{\kappa - \alpha}e^{- \alpha \tau}
- \frac{\alpha}{\kappa - \alpha}e^{-\kappa \tau} \\
e^{- \mu \tau}B'(\tau) =
-\frac{\alpha \kappa}{\kappa - \alpha}[e^{-(\mu + \alpha )\tau}
- e^{-(\mu + \kappa )\tau}]
\end{gather*}
With these choices and the identifications
$I = \phi$, $E = I^* - \phi$,
we may verify that the system (\ref{age}) reduces to
\begin{align*}
S' &= \Lambda (N) - \beta (N) SI - \mu S \\
E' &= \beta (N) SI - \kappa E \\
I' &= \kappa E - (\mu + \alpha )I \\
N' &= \Lambda (N) - (1 - f)\alpha I- \mu N,
\end{align*}
which is a standard $SEIR$ model.
For some diseases there is an asymptomatic period during which individuals
have some infectivity rather than an exposed period. If the infectivity during
this period is reduced by a factor $\epsilon$, then the model can be described
by the system
\begin{align*}
S' &= \Lambda (N) - \beta (N) S(I + \epsilon E) - \mu S \\
E' &= \beta (N) S(I + \epsilon E) - \kappa E \\
I' &= \kappa E - (\mu + \alpha )I \\
N' &= \Lambda (N) - (1 - f)\alpha I- \mu N,
\end{align*}
This may be considered as an age of infection model with the same identifications
of the variables and the same choice of $u(\tau), v(\tau)$ but with
$A(\tau) = \epsilon u(\tau) + v(\tau)$.
\section{Equilibria and the characteristic equation}
There is a disease-free equilibrium $S = N = K, \phi = 0$ of (\ref{age}).
Endemic equilibria are given by
\begin{gather*}
S \beta(N)\hat{A_\mu}(0) = 1 \\
\Lambda (N) = \mu N + (1 - f)(1 - \mu \hat{B_\mu}(0))S \beta (N) \phi \\
\Lambda (N) = \mu S + S \phi \beta (N)
\end{gather*}
If $f = 1$ the third condition gives
\[
\phi = \frac{\mu(K - \beta(K)}{\hat{A_\mu}(0)}
\]
and there is always an endemic equilibrium. If $f < 1$ the second of the
equilibrium conditions gives
\[
\phi = \frac{\hat{A_\mu}(0)}{M}[\Lambda(N) - \mu N]
\]
Now substitution of the first two equilibrium conditions into the third
gives an equilibrium condition for $N$, namely
\begin{equation} \label{N}
(1 - M)\Lambda(N) = \mu N - \frac{\mu M}{\beta (N)\hat{A_\mu}(0)}
\end{equation}
If $R_0 < 1$,
\[
C(N)\hat{A_\mu}(0) \leq C(K)\hat{A_\mu}(0) = R_0 < 1
\]
so that
\[
1 - \frac{M}{C(N)\hat{A_\mu}(0)} < 1 - M
\]
Then we must have $\Lambda (N) < \mu N$ at equilibrium. However, this would
contradict the demographic condition $\Lambda (N) > \mu N, 0 < N < K$.
This shows that if $R_0 < 1$ there is no
endemic equilibrium.
If $R_0 > 1$ for $N = 0$ the left side of
(\ref{N}) is non-negative while the right side is negative. For $N = K$
the left side of (\ref{N}) is $\mu K(1 - M)$ while the right side is
\[
\mu K - \frac{M\mu K}{R_0} > \mu K(1 - M)
\]
This shows that there is an endemic equilibrium solution for $N$.
The linearization of (\ref{age}) at
an equilibrium $(S, N, \phi)$ is
\begin{gather*}
x' = -[\mu + \phi \beta(N)]x + [\Lambda'(N) - S \phi \beta'(N)]y - S\beta(N)z \\
\begin{aligned}
y' &= [\Lambda'(N) - \mu]y + (1 - f)\int_0^\infty e^{-\mu\tau}
B(\tau)[\phi\beta(N)x(t-\tau) \\
&\quad + S\phi \beta'(N)y(t-\tau) + S \beta(N)z(t - \tau)]d\tau
\end{aligned}\\
z(t) = \int_0^\infty A_\mu(\tau)[\phi\beta(N)x(t-\tau)
+ S\phi \beta'(N)y(t-\tau) + S \beta(N)z(t - \tau)]d\tau
\end{gather*}
The condition that this linearization have solutions which are constant
multiples of $e^{-\lambda \tau}$ is that $\lambda$ satisfies a
characteristic equation. The characteristic equation at an equilibrium
$(S, \phi, N)$ is
\begin{align*}
&\det \begin{bmatrix}
-[\lambda + \mu + \phi \beta(N)] & [\Lambda'(N) - S \phi \beta'(N)] & S\beta(N) \\
-\phi \beta(N)Q(\lambda) & -[\lambda - \Lambda'(N) + \mu]
- S\phi \beta'(N)Q(\lambda) &
- S\phi \beta(N)Q(\lambda) \\
\phi \beta(N)\hat{A_\mu}(\lambda) & S\phi \beta'(N)\hat{A_\mu}(\lambda) &
S\beta(N)\hat{A_\mu}(\lambda) - 1
\end{bmatrix}\\
& =0\,,
\end{align*}
where
\begin{gather*}
\hat{A_\mu}(\lambda) = \int_0^\infty e^{-\lambda \tau}A_\mu(\tau)d\tau \,,\quad
\hat{B_\mu}(\lambda) = \int_0^\infty e^{-\lambda \tau}B_\mu(\tau)d\tau \,,\\
Q(\lambda) = (1 - f)[1 - (\lambda + \mu)\hat{B_\mu}(\lambda)]\,.
\end{gather*}
This reduces to
\begin{equation} \label{char}
\begin{aligned}
&S \beta(N) \hat{A_\mu}(\lambda) + (1 - f) \phi S \beta' (N) \hat{B_\mu}(\lambda)
\\
&= 1 + \frac{f \phi \beta(N)}{\lambda + \mu}
+ \frac{(1 - f) \phi P}{\lambda + \mu - \Lambda'(N)}
\cdot [1 - \Lambda'(N)\hat{B_\mu}(\lambda)]
\end{aligned}
\end{equation}
where $P = \beta(N) + S \beta'(N) \geq 0$.
The characteristic equation for a model consisting of a system of ordinary
differential equations is a polynomial equation. Now we have a transcendental
characteristic equation, but there is a basic theorem that if all roots of the
characteristic equation at an equilibrium have negative real part (that is,
if $\Re \lambda < 0$, where $\Re$ denotes the real part, for every root $\lambda$
of the characteristic equation) then the
equilibrium is asymptotically stable \cite{W85}.
At the disease-free equilibrium $ S = N = K, \phi = 0$ the characteristic
equation is
\[
K\beta(K)\hat{A_\mu}(\lambda) = 1\,.
\]
Since the absolute value of the left side of this equation is no greater than
$K \beta (K)\hat{A_\mu}(0)$ if $\Re \lambda \geq 0$
the disease-free equilibrium is asymptotically stable if and only if
\[
R_0 = K \beta (K)\hat{A_\mu}(0) < 1\,.
\]
\section{The Endemic Equilibrium}
In the analysis of the characteristic equation (\ref{char}) it is helpful to make
use of the following elementary result:
\begin{quote}
If $|P(\lambda)| \leq 1$, $\Re g(\lambda) > 0$ for $\Re \lambda \geq 0$,
then all roots of the characteristic equation
$P(\lambda) = 1 + g(\lambda)$
satisfies $\Re \lambda < 0$.
\end{quote}
To prove this result, we observe that if $\Re \lambda \geq 0$ the left side of
the characteristic equation has absolute value at most $1$ while the right
side has absolute value greater than $1$.
If $f = 1$, the characteristic equation reduces to
\[
S \beta(N) \hat{A_\mu}(\lambda) = 1 + \frac{\phi \beta(N)}{\lambda + \mu}\,.
\]
The term
\[
\frac{f \phi \beta(N)}{\lambda + \mu}
\]
in (\ref{char}) has positive real part if $\Re \lambda \geq 0.$ It follows from
the lemma that all roots satisfy $\Re \lambda < 0$, so that the endemic
equilibrium is asymptotically stable.
Thus all roots of the characteristic equation (\ref{char}) have negative real part
if $f = 1$.
The analysis if $f < 1$ is more difficult.
The roots of the characteristic equation depend continuously on the
parameters of the equation. In order to have a root with $\Re \lambda \geq 0$ there
must be parameter values for which either there is a root at ''infinity", or there is
a root $\lambda = 0$ or there is a pair of pure imaginary roots
$\lambda = \pm \, iy, y> 0$. Since the left side of (\ref{char}) approaches $0$
while the right side approaches $1$ as $\lambda \to \infty, \Re \lambda \geq 0$,
it is not possible for a root to appear at ``infinity". For $\lambda = 0$, since
$S \beta(N)\hat{A_\mu}(0) = 1$ and $\beta'(N) \leq 0$ the left side of (\ref{char})
is less than $1$ at $\lambda = 0$, while the right side is greater than $1$ since
\[
1 - \Lambda'(N)\hat{B_\mu}(0) > 1 - \Lambda'(N)/\mu > 0
\]
if $\Lambda'(N) < \mu$. This shows that $\lambda = 0$ is not a root of
(\ref{char}), and therefore that all roots satisfy $\Re \lambda < 0$ unless there is a
pair of roots $\lambda = \pm \,iy, y> 0$. According to the Hopf bifurcation theorem
\cite{H42} a pair of roots $\lambda = \pm \, iy, y> 0$ indicates that the system
(\ref{age}) has an asymptotically stable periodic solution and there are sustained
oscillations of the system.
A somewhat complicated calculation using the fact that since $B_\mu (\tau)$
is monotone non-increasing,
\[
\int_0^\infty B_\mu (\tau) \sin y\tau dy \geq 0
\]
for $0 \leq y < \infty$
shows that the term
\[
\frac{(1 - f) \phi P}{\lambda + \mu - \Lambda'(N)}
\cdot [1 - \Lambda'(N)\hat{B_\mu}(\lambda)]
\]
in (\ref{char}) has positive real part at least if
$- \mu \leq \Lambda'(N) \leq \mu$.
Therefore, if $- \mu \leq \Lambda'(N) \leq \mu$, instability of the endemic equilibrium
is possible only if the term
\[
(1 - f) \phi S \beta' (N) \hat{B_\mu}(iy)
\]
in (\ref{char}) has negative real part for some $y > 0$. This is not possible
with bilinear incidence, since $\beta'(N) = 0$; thus with bilinear incidence
the endemic equilibrium of (\ref{age}) is always asymptotically stable. Since
$\beta'(N) \leq 0$, instability requires
\[
\Re \hat{B_\mu}(iy) = \int_0^\infty B_\mu (\tau) \cos y\tau d\tau < 0
\]
for some $y > 0.$ If the function $B(\tau)$ is non-increasing and convex,
that is, if $B'(\tau) \leq 0, B''(\tau) \geq 0$, then it is easy to show
using integration by parts that
\[
\int_0^\infty B_\mu (\tau) \cos y\tau d\tau \geq 0
\]
for $0 < y < \infty$. Thus if $B(\tau)$ is convex, which is satisfied for example,
by the choice
\[
B(\tau) = e^{- \alpha \tau}
\]
the endemic equilibrium of (\ref{age}) is asymptotically stable if
$- \mu \leq \Lambda'(N) \leq \mu$.
There are certainly less restrictive conditions which guarantee asymptotic
stability. However, examples have been given of instability, even with $f = 0,
\Lambda'(N) = 0$, where constant infectivity would have produced asymptotic
stability \cite{B96, TC89, TC93}. These results indicate
that concentration of infectivity early in the infected period is conducive
to such instability. In these examples, the instability arises because a root
of the characteristic equation crosses the imaginary axis as parameters of the
model change, giving a pure imaginary root of the characteristic equation.
This translates into oscillatory solutions of the model. Thus infectivity which
depends on infection age can cause instability and sustained oscillations.
\section{An $SI^*S$ Model}
To formulate an $SI^*S$ age of infection model we need only take the $SI^*R$
age of infection model (\ref{age}) and move the recovery term from the
equation for $R$ (which was not listed explicitly in the model) to the equation for $S$.
We obtain the model
\begin{equation}
\begin{aligned}
S' &= \Lambda (N) - \mu S - \beta (N) S \phi - \\
&= f\int_0^\infty \beta (N(t - \tau))
S(t - \tau)\phi (t - \tau) e^{-\mu \tau}B'(\tau) d\tau \\
\phi(t) &= \int_0^\infty \beta (N(t - \tau)) S(t - \tau)\phi (t - \tau)
A_\mu(\tau) d\tau \\
N'(t) &= \Lambda (N) - \mu N
+ (1 - f)\int_0^\infty \beta (N(t - \tau)) S(t - \tau)\phi (t - \tau)
e^{-\mu \tau}B'(\tau) d\tau
\end{aligned} \label{age3}
\end{equation}
Although we will not carry out any analysis of this model, we state that
it may be attacked using the same approach as that used for (\ref{age}).
It may be shown that if
$R_0 = K \beta(K)\hat{A_\mu}(0) < 1$ the disease-free equilibrium is
asymptotically stable. If $R_0 > 1$ there is an endemic equilibrium and the
characteristic equation at this equilibrium is
\begin{equation} \label{char2}
\begin{aligned}
&S \beta(N) \hat{A_\mu}(\lambda) + (1 - f) \phi S \beta' (N) \hat{B_\mu}(\lambda)\\
&= 1 + f \phi \beta(N)\hat{B_\mu}(\lambda)
+ \frac{(1 - f) \phi P}{\lambda + \mu - \Lambda'(N)}
\cdot [1 - \Lambda'(N)\hat{B_\mu}(\lambda)]\,,
\end{aligned}
\end{equation}
where $P = \beta(N) + S \beta'(N) \geq 0$.
Many diseases, including most strains of influenza, impart only temporary immunity
against reinfection on recovery. Such disease may be described by $SI^*S$ age of
infection models, thinking of the infected class $I^*$ as comprised of the infective
class $I$ together with the recovered and immune class $R$. In this way, members of
$R$ neither spread or acquire infection. We assume that immunity is lost at a
proportional rate $\kappa$.
We let $u(\tau)$ be the fraction of infected members with infection age $\tau$
who are infective if alive and $v(\tau)$ the fraction of
infected members who are not recovered and still immune if alive. Then the
fraction becoming immune at infection age $\tau$ if alive is $\alpha u(\tau)$,
and we have
\begin{equation} \label{exp2}
\begin{gathered}
u'(\tau) = - \alpha u(\tau), \quad u(0) = 1 \\
v'(\tau) = \alpha u(\tau) - \kappa v(\tau) \quad v(0) = 0
\end{gathered}
\end{equation}
These equations are the same as (\ref{exp}) obtained in formulating the $SEIR$
model with $\alpha$ and $\kappa$ interchanged. Thus we may solve to obtain
\begin{gather*}
u(\tau) = e^{-\alpha \tau} \\
v(\tau) = \frac{\alpha}{\kappa - \alpha}[e^{-\alpha \tau} - e^{- \kappa \tau}]
\end{gather*}
We take $B(\tau) = u(\tau) + v(\tau)$, $A(\tau) = u(\tau)$. Then if we define
$I = \phi, R = I^* - \phi$, the model (\ref{age3}) is equivalent to the system
\begin{align*}
S' &= \Lambda (N) - \beta (N) SI - \mu S + \kappa R \\
I' &= \beta (N) SI - (\mu + \alpha) I \\
R' &= f\alpha E - (\mu + \kappa )R \\
N' &= \Lambda (N) - (1 - f)\alpha I- \mu N,
\end{align*}
which is a standard $SIRS$ model.
If we assume that, instead of an exponentially distributed immune period,
that there is an immune period of fixed length $\omega$ we would again
obtain $u(\tau) = e^{-\alpha \tau}$, but now we may calculate that
\[
v(\tau) = 1 - e^{-\alpha \tau}, (\tau \leq \omega),
\quad v(\tau) = e^{-\alpha \tau}(e^{\alpha \omega} - 1), (\tau > \omega).
\]
To obtain this, we note that
\[
v'(\tau) = \alpha u(\tau), 0 \leq \omega, \quad v'(\tau) =
\alpha u(\tau) - \alpha u(\tau - \omega), \tau > \omega
\]
From these we may calculate $A(\tau), B(\tau)$ for an $SI^*S$ model. Since it
is known that the endemic equilibrium for an $SIRS$ model with a fixed removed
period can be unstable \cite{HSV81},
this shows that (\ref{char2}) may have roots with
non-negative real part and the endemic equilibrium of an $SI^*S$ age of
infection model is not necessarily asymptotically stable.
\section{Discussion}
We have formulated some general age of infection epidemiological models and set up
their equilibrium analysis. Compartmental models which include exposed periods,
temporary immunity, and other compartments, can be formulated as age of
infection models.
The $SI^*R$ age of infection model is actually a special case of the $SI^*S$
age of infection model. We may view the class $R$ as still infected but
having no infectivity, so that $v(\tau) = 0$. The underlying idea is that in
infection age models we divide the population into members who may become
infected and members who can not become infected, either because they are
already infected or because they are immune. Thus, we may view the $SI^*S$
model as general. If we could carry out a complete analysis
of the corresponding characteristic equation we could use it as the basis of a
general theory. However, since the characteristic equation analysis is not yet
complete there are many open questions whose answers would provide useful
insights into general model behaviour.
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\end{document}