Analysis of an SIS epidemic model with treatment

Infectious diseases have tremendous influence on human life and will bring huge panic and disaster to mankind once out of control. Every year millions of human beings suffer from or die of various infectious diseases. In order to predict the spreading of infectious diseases, many epidemic models have been proposed and analyzed in recent years (see [1–13]). Some new conditions should be considered into SIS model to extend the results.

Li et al. (see [13]) studied an SIS model with bilinear incidence rate $\beta SI$ and treatment. The model takes into account the medical conditions. The recovery of the infected rate is divided into natural and unnatural recovery rates. Because of the medical conditions, when the number of infected persons reaches a certain amount $I_0$, the unnatural recovery rate will be a fixed value $\delta I_0$. The study of this model should be divided into two cases to discuss with $I\le I_0$ and $I>I_0$. In this paper, we study an SIS model with saturated incidence rate $\frac{\beta SI}{1+\alpha S}$ and treatment, and we extend some recent results.

where $T(I)=\{\begin{array}{ll}\delta I,& \text{if }0\le I\le I_0,\\ k,& \text{if }I\ge I_0\end{array}$ ($k=\delta I_0$) is the rate at which infected individuals are treated; A is the recruitment rate of individuals (including newborns and immigrants) into the susceptible population; $\frac{\beta SI}{1+\alpha S}$ is the nonlinear incidence rate; d is the natural death rate; γ is the rate at which infected individuals are recovered; ε is the disease-related death rate, A, d, γ, δ, ε, α are all positive numbers.

We easily see that model (1.1) has a disease-free equilibrium $P_0(\frac{A}{d},0)$.

Let $R_0=\frac{\beta A}{(d+A\alpha )(d+\epsilon +\gamma +\delta )}$. We study equation (2.5) as follows.

So this case need not be considered.

Therefore model (1.1) has a disease-free equilibrium $P_0(\frac{A}{d},0)$ and has an endemic equilibrium $P_{\ast }(S_{\ast },I_{\ast })$ except the disease-free equilibrium $P_0(\frac{A}{d},0)$ when $1<R_0\le n_0$.

Let $b=(A\alpha +d)(d+\epsilon +\gamma )-\beta A-k\alpha (d+\epsilon )$. We study equation (2.6) as follows.

So this case need not be considered.

Then $-b+2(d+\epsilon )[\alpha (d+\epsilon +\gamma )-\beta ]I_0<0$.

If $\beta >\alpha (d+\epsilon +\gamma )$, it is easy to see that (2.6) has no positive root if $b\ge 0$.

At the same time, $b<0$ holds if and only if $R_0>1-\frac{\delta (A\alpha +d)+k\alpha (d+\epsilon )}{(A\alpha +d)(d+\epsilon +\delta +\gamma )}$.

It is easy to see that $n_8<n_7$, which implies that (2.6) has two positive equilibrium points $P_2(S_2,I_2)$, $P_3(S_3,I_3)$ if $R_0<n_8$, (2.6) has only one positive equilibrium point $P_2(S_2,I_2)$ if $n_8<R_0<n_7$, (2.6) has no positive equilibrium point if $R_0\ge n_7$.

Therefore, if $n_9<R_0<n_3$, $I_2>I_0$ holds.

we get $R_0\le n_9$ or $R_0>max(n_3,n_9)$.

From the above discussion, we get the following conclusions.

Theorem 2.1 If $R_0<1$, model (1.2) has only one disease-free equilibrium $P_0(\frac{A}{d},0)$; if $R_0>1$, model (1.2) has a unique endemic equilibrium $P_{\ast }(S_{\ast },I_{\ast })$ except the disease-free equilibrium $P_0(\frac{A}{d},0)$; if $1<R_0<n_0$, $P_{\ast }(S_{\ast },I_{\ast })$ is a unique endemic equilibrium of model (1.1).

Theorem 2.2 If $\beta <\alpha (d+\epsilon +\gamma )$, then $P_1(S_1,I_1)$ is a unique endemic equilibrium of model (1.3) if $R_0\le n_1$; $P_1(S_1,I_1)$ is a unique endemic equilibrium of model (1.1) if $R_0\le n_1$ and $R_0\le n_2$.

If $\beta >\alpha (d+\epsilon +\gamma )$, model (1.3) has two positive equilibrium points $P_2(S_2,I_2)$, $P_3(S_3,I_3)$ if $R_0<n_8$; model (1.3) has only one positive equilibrium point $P_2(S_2,I_2)$ if $n_8<R_0<n_7$; model (1.3) has no positive point if $R_0\ge n_7$; $P_2(S_2,I_2)$ is an endemic equilibrium of model (1.1) if $n_9<R_0<n_3$; $P_3(S_3,I_3)$ is an endemic equilibrium of model (1.1) if $R_0<n_9$ or $n_9<R_0<n_3$.

If $\beta =\alpha (d+\epsilon +\gamma )$, model (1.3) has no endemic equilibrium.

Theorem 3.1 The disease-free equilibrium $P_0(\frac{A}{d},0)$ is stable if $R_0<1$ and is a saddle point if $R_0>1$; the endemic equilibrium $P_{\ast }(S_{\ast },I_{\ast })$ is a stable node if it exists; the endemic equilibrium $P_1(S_1,I_1)$ is a stable node if it exists; if the endemic equilibrium points $P_2(S_2,I_2)$, $P_3(S_3,I_3)$ exist, then $P_2(S_2,I_2)$ is a stable node of model (1.1) if $\frac{-b-\sqrt{{\mathrm{\Delta }}_2}}{2(d+\alpha )[\beta -\alpha (d+\alpha +\gamma )]}{D}_1+D_2>0$ and $P_3(S_3,I_3)$ is a stable node of model (1.1) if $\frac{-b+\sqrt{{\mathrm{\Delta }}_2}}{2(d+\alpha )[\beta -\alpha (d+\alpha +\gamma )]}{D}_1+D_2>0$.

Thus, $P_0(\frac{A}{d},0)$ is a stable node if $R_0<1$, and is a saddle point if $R_0>1$.

So, $P_{\ast }(S_{\ast },I_{\ast })$ is a stable node if it exists.

If $\beta <\alpha (d+\epsilon +\gamma )$, $P_1(S_1,I_1)$ does not exist, then $detJ(P_1)>0$.

then $P_1(S_1,I_1)$ is a stable node if it exists.

If $\frac{-b-\sqrt{{\mathrm{\Delta }}_2}}{2(d+\alpha )[\beta -\alpha (d+\alpha +\gamma )]}{D}_1+D_2>0$, $detJ(P_2)>0$ holds, then $P_2(S_2,I_2)$ is a stable node if $\beta >\alpha (d+\epsilon +\gamma )$ and $\frac{-b-\sqrt{{\mathrm{\Delta }}_2}}{2(d+\alpha )[\beta -\alpha (d+\alpha +\gamma )]}{D}_1+D_2>0$. Similarly, $P_3(S_3,I_3)$ is a stable node if $\beta >\alpha (d+\epsilon +\gamma )$ and $\frac{-b+\sqrt{{\mathrm{\Delta }}_2}}{2(d+\alpha )[\beta -\alpha (d+\alpha +\gamma )]}{D}_1+D_2>0$. This completes the proof. □

Theorem 3.2 If $\delta <d$, there is no limit cycle of model (1.1).

Thus $\frac{\partial (DP)}{\partial S}+\frac{\partial (DQ)}{\partial I}<0$ if $\delta <d$.

Then there is no limit cycle of model (1.1) if $\delta <d$. This completes the proof. □

Theorem 3.3 There is no limit cycle of model (1.1) if $\beta \le \alpha (2d+\epsilon +\gamma )$.

Thus $\frac{\partial (DP)}{\partial S}+\frac{\partial (DQ)}{\partial I}<0$ if $\beta \le \alpha (2d+\epsilon +\gamma )$.

Then there is no limit cycle of model (1.1) if $\beta \le \alpha (2d+\epsilon +\gamma )$. This completes the proof. □

With different A, d, γ, δ, ε, α, it is easy to test and verify the above results, so numerical simulation is omitted. In this paper, we study an SIS model with saturated incidence rate $\frac{\beta SI}{1+\alpha S}$ and treatment. We get some relatively complex conclusions by stability theory and qualitative theory of differential equations. These conclusions will help policy makers to make decisions.