Media alert in an SIS epidemic model with logistic growth

Figures & data

Figure 1. The existence of the endemic equilibria of model (3(3) $\begin{array}{rl}\frac{\mathrm{d}S}{\mathrm{d}t}& =rS\left(1-\frac{S}{a}\right)-\beta \left(I\right)IS+\gamma I,\\ \frac{\mathrm{d}I}{\mathrm{d}t}& =\beta \left(I\right)IS-(d+ϵ+\gamma )I,\end{array}$(3) ) in the case of p>1. There are equilibria $E_{\ast }$ and $E_{\mathbf{\text{e}}}$ on the curve $L_1$, $E_{\ast }$ and $E_1$ on $L_2$, three equilibria $E_{\ast }$, $E_2$ and $E_3$ in the region $Q_2$, a unique equilibrium $E_1$ in the region $Q_1$, a unique equilibrium $E_{\ast }$ in the region $Q_3$, and no endemic equilibrium in $Q_4$. See the content for the definitions of these regions.

Figure 2. Phase plot of I verses S showing that two stable endemic equilibria $E_{\ast }$, $E_1$ coexist. Here, we fix $(a,b,\beta ,d,\gamma ,ϵ)=(600,1.9237\times {10}^{-3},0.0139,\frac{1}{75},0.0196,2.7397)$, and set $I_c=45>I_c^0=21$.

Figure 3. Phase plot of I verses S showing that bi-stability occurs for $I_c>I_c^0$ and $a_1<a<a_2$. Here, $E_{\ast }$ and $E_3$ are locally stable while $E_2$ is a saddle point. $I_c=50>I_c^0=21$ and other parameters take the same values as in Figure 2.

Figure 4. A limit cycle intersects with the line $I=I_c$ at $C_1$ and $C_2$.

Figure 5. The number of infected cases is stabilized at decreased levels as either p increases or $I_c$ decreases.