# 致病和治病同时传播的传染病

### 两种传染病抵消机制的简化描述

\begin{align}
q^{\left(1\right)}=p\left(1_{j},t\right) = \frac{\eta^{\left(1\right)}_{j} \left(t\right) – \eta^{\left(-1\right)}_{j} \left(t\right)}{\eta^{\left(1\right)}_{j} \left(t\right) + \eta^{\left(-1\right)}_{j} \left(t\right)}\theta\left(\eta^{\left(1\right)}_{j} \left(t\right) – \eta^{\left(-1\right)}_{j} \left(t\right)\right) \notag \\
q^{\left(-1\right)}=p\left(-1_{j},t\right) = \frac{\eta^{\left(-1\right)}_{j} \left(t\right) – \eta^{\left(1\right)}_{j} \left(t\right)}{\eta^{\left(1\right)}_{j} \left(t\right) + \eta^{\left(-1\right)}_{j} \left(t\right)}\theta\left(\eta^{\left(-1\right)}_{j} \left(t\right) – \eta^{\left(1\right)}_{j} \left(t\right)\right) \notag \\
p\left(0_{j},t\right) = 1-\frac{\left|\eta^{\left(1\right)}_{j} \left(t\right) – \eta^{\left(-1\right)}_{j} \left(t\right)\right|}{\eta^{\left(1\right)}_{j} \left(t\right) + \eta^{\left(-1\right)}_{j} \left(t\right)} = 1- q^{\left(1\right)} -q^{\left(-1\right)}
\end{align}

1. 如果$$k_{j}=0$$，则没有影响
2. 如果$$k_{j}=1$$（这个时候可以约定$$\eta^{\left(1\right)}_{j}=1,\eta^{\left(-1\right)}_{j}=0$$），则采用乘性传播
3. \begin{align}
\eta^{\left(1\right)}_{i}\left(t\right) = \eta^{\left(1\right)}_{i}\left(t-1\right) + \omega^{\left(1\right),j}_{i}\eta^{\left(1\right)}_{j}\left(t-1\right)
\end{align}
或者加性传播
\begin{align}
\eta^{\left(1\right)}_{i}\left(t\right) = \eta^{\left(1\right)}_{i}\left(t-1\right) + \omega^{\left(1\right),j}_{i}
\end{align}

4. 如果$$k_{j}=-1$$（这个时候可以约定$$\eta^{\left(1\right)}_{j}=0,\eta^{\left(-1\right)}_{j}=1$$），则采用乘性传播
5. \begin{align}
\eta^{\left(-1\right)}_{i}\left(t\right) = \eta^{\left(-1\right)}_{i}\left(t-1\right) + \omega^{\left(-1\right),j}_{i}\eta^{\left(-1\right)}_{j}\left(t-1\right)
\end{align}
或者加性传播
\begin{align}
\eta^{\left(1\right)}_{i}\left(t\right) = \eta^{\left(1\right)}_{i}\left(t-1\right) + \omega^{\left(-1\right),j}_{i}
\end{align}

\begin{align}
\eta^{\left(k_{j}\right)}_{i}\left(t\right) = \eta^{\left(k_{j}\right)}_{i}\left(t-1\right) + \omega^{\left(k_{j}\right),j}_{i}\eta^{\left(k_{j}\right)}_{j}\left(t-1\right)\left(k^{j}\right)^{2}
\end{align}

\begin{align}
\eta^{\left(k_{j}\right)}_{i}\left(t\right) = \eta^{\left(k_{j}\right)}_{i}\left(t-1\right) + \omega^{\left(k_{j}\right),j}_{i}\left(k^{j}\right)^{2}
\end{align}