Program 4.4: Multi-host SIR model
This is an example of a multi-host SIR model where there are two hosts (humans and mosquitoes). Transmissions within human or mosquito populations is also possible. It is adapted from a MATLAB code example in Keeling & Rohani.
$\frac{dX_H}{dt}=\nu_H-r(T_{HM}Y_M+T_{HH}Y_H)X_H-\mu_HX_H$
$\frac{dX_M}{dt}=\nu_M-r(T_{MH}Y_H+T_{MM}Y_M)X_M-\mu_MX_M$
$\frac{dY_H}{dt}=r(T_{HM}Y_M+T_{HH}Y_H)X_H-\mu_HY_H-\gamma_HY_H$
$\frac{dY_M}{dt}=r(T_{MH}Y_H+T_{MM}Y_M)X_M-\mu_MY_M-\gamma_MY_M$
where:
$X_H$ & $X_M$ are the human and mosquito susceptible populations and $Y_H$ & $Y_M$ are the infected ones. $\nu_H$ & $\nu_M$ are the birth rates of humans and mosquitoes. $r$ is the mosquito biting rate of humans. $T$ is a 2x2 matrix of transmission probabilities between and within human($H$) and mosquito($M$) populations. $\mu_H$ & $\mu_M$ are the per capita death rate of humans and mosquitoes. $\gamma_H$ & $\gamma_M$ are the recovery rates of humans and mosquitoes.