# Tempo-spatial Epidemic Models¶

We will use the Hantavirus as an example [Abramson2002].

The vector of Hantavirus is mice. The population of mice $$M$$ has two components, the infected population $$M_I$$ and the susceptible population $$M_S$$. We take the SI model with demography,

(46)$\begin{split}\frac{dM_S}{dt} &= \lambda M - \mu M_S - a M_S M_I, \\ \frac{dM_I}{dt} &= -\mu M_I + a M_S M_I,\end{split}$

where $$M = M_I + M_S$$, $$\lambda$$ is the natural birth rate per population, $$\mu$$ is the natural death rate per population, $$a$$ is the rate of the mouse being infect on contact.

Summing up the two equations, we get the equation for $$M$$,

$\frac{dM}{dt} = (\lambda - \mu) M,$

which has an exponential solution

$M = M_0 e^{(\lambda -\mu)t}.$

This is a trivial solution. As the birth rate and death rate becomes the same, the population becomes stable.

However, the population should also be limited by the natural resources they consume,

$\frac{dM}{dt} = (\lambda - \mu) M - f(M, t) M,$

where $$f(M, t)M$$ describes the competitions for resources.

$$f(M, t)$$

Suppose we have a bunch of resources $$K$$ and each mouse requires $$k$$ to survive on average. The more mice we have, the more competitive,

$f(M,t) = \frac{kM}{K}.$

Suppose we have equal birth rate and death rate, the population is governed by

$\frac{dM}{dt} = \frac{kM}{K}M.$

For simplifity, we define $$\kappa=k/K$$,

$\frac{dM}{dt} = (\lambda - \mu) M - \frac{M}{\kappa} M.$

Following [Abramson2002], we introduce the competition term

$\begin{split}\frac{dM_S}{dt} &= \lambda M - \mu M_S - a M_S M_I {\color{blue}- \frac{M}{\kappa} M_S}, \\ \frac{dM_I}{dt} &= -\mu M_I + a M_S M_I {\color{blue}- \frac{M}{\kappa} M_I}.\end{split}$

The vector, deer mice, is quite mobile. The spatial distribution becomes important. To describe the spatial component, we add a diffusion term

$\begin{split}\frac{\partial M_S}{\partial t} &= \lambda M - \mu M_S - a M_S M_I {\color{blue}- \frac{M}{\kappa} M_S} {\color{red}+D_S \nabla^2M_S}, \\ \frac{dM_I}{dt} &= -\mu M_I + a M_S M_I {\color{blue}- \frac{M}{\kappa} M_I} {\color{red}+D_I \nabla^2M_I}.\end{split}$

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