Epidemic process

A random process (cf. Stochastic process) that serves as a mathematical model of the spread of some epidemy. One of the simplest such models can be described as a continuous-time Markov process whose states at the moment $t$ are the number ${\mu }_1(t)$ of sick persons and the number ${\mu }_2(t)$ of exposed persons. If ${\mu }_1(t)=m$ and ${\mu }_2(t)=n$, then at the time $t$, $t+\mathrm{\Delta }t$, $\mathrm{\Delta }t\to 0$, the transition probability is determined as follows: $(m,n)\to (m+1,n-1)$ with probability ${\lambda }_{mn}\mathrm{\Delta }=O(\mathrm{\Delta }t)$; $(m,n)\to (m-1,n)$ with probability $\mu m\mathrm{\Delta }t+O(\mathrm{\Delta }t)$. In this case the generating function

$$F(t;x,y)=\mathsf{E}{x}^{{\mu }_1(t)}{y}^{{\mu }_2(t)}$$

satisfies the differential equation

$$\frac{\partial F}{\partial t}=\lambda (x^2-xy)\frac{{\partial }^{2}F}{\partial x\partial y}+\mu (1-x)\frac{\partial F}{\partial x}.$$

Comments[edit]

References[edit]