Category Archives: Mathematical Biology

Simple Epidemics: Stochastic Model 1

Suppose that initially there are $n$ susceptibles and one infective. Let $X(t)$ denote a random variable which represents that number of susceptibles still uninfected at time $t$. The probability that $X(t)$ the value $r$ is $p_r(t)$. At time $t$, there are $X(t)$ susceptibles and $n-X(t)+1$ infectives. Assume that the chance of a new infection in a short time interval is proportional to the product of the number of susceptibles, the number of infectives and the length of the interval. Then the chance of an infection in $\Delta t$ is given by $\beta X(t)(n-X(t)+1)\Delta t$, where $\beta$ is the contact rate. For simplicity, we change the time scale to $\tau=\beta t$. The chance then becomes $X(n-X+1)\Delta\tau$. Denote by $p_r(\tau)$ the probability that there are still $r$ susceptibles remaining uninfected at $\tau$. There are two possibilities that can occur at time $\tau+\Delta\tau$: There are either $r+1$ susceptibles at $\tau$ followed by a new infection with probability $(r+1)(n-r)\Delta\tau$ , or $r$ susceptibles at $\tau$ followed by no infection with probability $1-r(n-r+1)\Delta\tau$. Thus $p_r(\tau+\Delta\tau)$, the probability of $r$ susceptibles remaining at time $\tau+\Delta\tau$ is given by $$p_r(\tau+\Delta\tau)=(r+1)(n-r)\Delta\tau p_{r+1}(\tau)+\{1-r(n-r+1)\Delta\tau\}p_r(\tau)$$ from which we obtain the differential-difference equation \begin{align*}\frac{dp_r(\tau)}{d\tau}&=\lim_{\Delta\tau\to 0}\frac{p_r(\tau+\Delta\tau)-p_r(\tau)}{\Delta\tau}\\&=(r+1)(n-r)p_{r+1}(\tau)-r(n-r+1)p_r(\tau)\end{align*} where $0\leq r\leq n-1$. For $r=n$ at $\tau+\Delta\tau$, there are $n$ susceptibles at $\tau$ followed by no new infection with probability $1-n\Delta\tau$. Thus we have $$p_n(\tau+\Delta\tau)=(1-n\Delta\tau)p_n(\tau)$$ and $$\frac{dp_n}{d\tau}=-np_n(\tau)$$ At $\tau=0$, there is only one infective, so we have the initial condition $p_n(0)=1$. We now have the system of differential-difference equations \begin{equation}\label{eq:stocepidemic}\frac{dp_r(\tau)}{d\tau}=(r+1)(n-r)p_{r+1}(\tau)-r(n-r+1)p_r,\ 0\leq r\leq n\end{equation} with initial condition $p_n(0)=1$.

One may attempt to solve the system \eqref{eq:stocepidemic} of differential-difference equations successively, similarly to that of the Poisson distribution (see here). The first three solutions are given by \begin{align*}p_n(\tau)&=e^{-n\tau}\\p_{n-1}(\tau)&=\frac{n}{n-1}[1-e^{-(n-1)\tau}]e^{-(n-1)\tau}\\p_{n-1}(\tau)&=\frac{2n}{2n-5}e^{-(n-1)\tau}-\frac{2n}{n-4}e^{-2(n-1)\tau}+\frac{2n(n-1)}{(n-4)(2n-5)}e^{-3(n-2)\tau}\end{align*} It is already difficult to see any recognizable pattern that may lead us to a general form of the solution. There are alternative ways to solve \eqref{eq:stocepidemic}, one of which is by using a generating function (see here). Let $P(x,r)=\sum_{r=0}^np_r(\tau)x^r$. Multiplying \eqref{eq:stocepidemic} by $x^r$ and summing the result over $r$ from $r=0$ to $r=\infty$, we obtain the partial differential equation $$\frac{\partial P}{\partial\tau}=(1-x)\left\{n\frac{\partial P}{\partial x}-x\frac{\partial^2 P}{\partial x^2}\right\}$$ with initial condition $P(x,0)=x^n$.

References:

  1. Norman T. J. Bailey, A Simple Stochastic Epidemic, Biometrika, Vol. 37, No. 3/4 (Dec., 1950), 193-202.
  2. The Mathematical Theory of Infectious Diseases and Its Applications, Norman T. J. Bailey, Second Edition, Charles Griffin & Company LTD, 1975.

Simple Epidemics: Deterministic Model

Suppose that we have a homogeneously mixing group of individuals of total size $n+1$ and that the epidemics is started off at $t=0$ by one individual becoming infectious. The remaining $n$ individuals are susceptible. Denote by $x$ and $y$ the numbers of susceptibles and infectives, respectively. Then $x+y=n+1$. Also suppose that the rate of occurrence of new infections is proportional to the number of infectives as well the number of susceptibles. Then we obtain the following differential equation \begin{equation}\label{eq:infectrate}\frac{dx}{dt}=-\beta xy=-\beta x(n-x+1)\end{equation} where $\beta$ is the constant of proportionality called the infection-rate. Let $\tau=\beta t$. \eqref{eq:infectrate} becomes \begin{equation}\label{eq:infectrate2}\frac{dx}{d\tau}=-x(n-x+1)\end{equation} With initial condition $x(0)=n$, the solution of \eqref{eq:infectrate2} is given by $$x(\tau)=\frac{n(n+1)}{n+e^{(n+1)\tau}}$$ The number of infectives at $\tau$ is then given by $$y(\tau)=\frac{n+1}{1+ne^{-(n+1)\tau}}$$ The rate at which new infectives accrue $$w(\tau)=\frac{dy}{d\tau}$$ is called the epidemic curve. It follows from \eqref{eq:infectrate2} that $$w(\tau)=-\frac{dx}{d\tau}=-xy=\frac{n(n+1)^2e^{(n+1)\tau}}{[n+e^{(n+1)\tau}]^2}$$ Using the standard argument from calculus, we find that $w(\tau)$ assumes its maximum when $x=y$ i.e. when the number of susceptibles and the number of infectives are the same and that happens when $\tau=\frac{\ln n}{n+1}$, and the maximum rate at which new infections accrue is $w=\frac{1}{4}(n+1)^2$.

Figure 1. Deterministic Epidemic Curve for n=20 (red) and for n=30 (blue).

As a reminder the model we discussed above assumed that the epidemic is started off by one individual (patient zero). Now, we want to generalized the model by assuming that the epidemic is started off by several infectious individuals. Let $x(0)=n$ and $y(0)=a$. Then the equation \eqref{eq:infectrate2} is modified to \begin{equation}\label{eq:infectrate3}\frac{dx}{dt}=-x(n-x+a)\end{equation} The solution of \eqref{eq:infectrate3} is $$x(\tau)=\frac{n(n+a)}{n+ae^{(n+a)\tau}}$$ The corresponding epidemic curve is given by $$w=\frac{an(n+a)^2e^{(n+a)\tau}}{[n+ae^{(n+a)\tau}]^2}$$ Let $W=\frac{w}{n}=\frac{a(n+a)^2e^{(n+a)\tau}}{[n+ae^{(n+a)\tau}]^2}$, $0\leq\tau<\infty$. Then \begin{align*}\int_0^\infty Wd\tau&=\int_{n+a}^\infty\frac{(n+a)}{u^2}du\ (u=n+ae^{(n+a)\tau})\\&=1\end{align*} So, $W$ can be interpreted as the probability density of a new infection occurring. The mean value of the variable $\tau$ is \begin{align*}\bar\tau&=\int_0^\infty\tau Wd\tau\\&=-\frac{1}{n}\int_0^\infty\tau\frac{dx}{d\tau}d\tau\\&=\frac{1}{n}\ln\left(1+\frac{n}{a}\right)\end{align*}

References:

  1. Norman T. J. Bailey, A Simple Stochastic Epidemic, Biometrika, Vol. 37, No. 3/4 (Dec., 1950), 193-202.
  2. The Mathematical Theory of Infectious Diseases and Its Applications, Norman T. J. Bailey, Second Edition, Charles Griffin & Company LTD, 1975.