An Infectious Disease Model With Media Coverage and Limited Medical Resources
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摘要:
该文建立和分析了一类具有媒体报道效应和有限医疗资源的传染病动力学模型，定义了疾病的基本再生数，分析了平衡点的存在性和稳定性，给出了系统发生前向分支、后向分支和Hopf分支的条件。通过数值模拟发现：提高媒体报道的信息覆盖率或医院对病人的最大容纳量，可以显著降低疾病流行的峰值或稳态时的感染人数；随着参数变化，系统不仅可能会产生后向分支或前向分支，还可能会出现鞍结点分支、Hopf 分支以及地方病平衡点稳定性随参数变化而变化等动力学行为。
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关键词:
- 媒体报道 /
- 有限医疗资源 /
- 后向分支 /
- Lyapunov 稳定
Abstract:A dynamical model for infectious disease with media coverage effects and limited medical resources was established and analyzed. The basic reproduction number of the disease was defined, the existence and stability of the equilibria were analyzed, and the conditions for the forward bifurcation, the backward bifurcation and the Hopf bifurcation to occur in the system were given. Numerical simulation results show that, increasing the media coverage rate or the maximum hospital capacity can significantly reduce the number of infections at the peak or in the steady state of the epidemic. With the variations of parameters, the system will exhibit dynamic behaviors including not only the backward bifurcation or the forward bifurcation, but also the saddle node bifurcation, the Hopf bifurcation, or the change of the endemic equilibrium point stability with parameters.
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Key words:
- limited medical resources /
- media coverage /
- backward bifurcation /
- Lyapunov stability
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图 1 当
$ q = 0.03,\; 0.08,\; 0.2\; $ 时，模型(2)解的长期行为Figure 1. For
$ q = 0.03,\; 0.08,\; 0.2 $ , the long-term behavior of the solution for model (2)
图 2 当
$K = 20,\; 30,\; 50$ 时，模型(2)解的长期行为注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同。
Figure 2. For
$ K = 20,\; 30,\; 50 $ , the long-term behavior of the solution for model (2)
图 3 当
$ \varLambda = 5 $ 或1时，模型(2)的非负平衡点随$ R_0 $ 变化的分支图：(a)$ \varLambda = 5 $ ； (b)$ \varLambda = 1 $ Figure 3. For
$ \varLambda = 5 $ or 1 , the bifurcation of nonnegative equilibria for model (2) with respective to$ R_0 $ : (a)$\varLambda = 5$ ; (b)$\varLambda = 1$ 
图 4 模型(2)解的最终性态对初值的依赖性 (双稳定现象)
Figure 4. Dependence of the final behavior of the solution for model (2) on the initial value (bistable phenomenon)
图 5 当
$R_{0}=1.3$ 时，模型(2)解的长期行为，其中$\varLambda=1,\; \mu=0.008,\; K=2,\; r=0.2,\; \gamma_1=0.06,\;\gamma_2=0.3$ ,$d_1=0.01,\;d_2=0.001,\; a=0.7,\; \tau=0.2,\; q=0.03$ ，初值为$(100,5,2,10)$ Figure 5. For
$R_{0}=1.3$ , the long-term behavior of the solution for model (2), where$\varLambda=1,\; \mu=0.008,\; K=2,\; r=0.2,\; \gamma_1=0.06,\;\gamma_2=0.3$ ,$d_1=0.01,\;d_2=0.001,\; a=0.7,\; \tau=0.2,\; q=0.03$ , and the initial value is$(100,5,2,10)$ -
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