Zhu Min, Guo Xiaofei, Lin Zhigui
School of Mathematical Science, Yangzhou University, Yangzhou 225002, China email:
Department of Mathematics, Anhui Normal University, Wuhu 241000, China email:
Math Biosci Eng. 2017;14(5-6):1565-1583. doi: 10.3934/mbe.2017081.
In this paper, a reaction-diffusion-advection SIR model for the transmission of the infectious disease is proposed and analyzed. The free boundaries are introduced to describe the spreading fronts of the disease. By exhibiting the basic reproduction number RDA0 for an associated model with Dirichlet boundary condition, we introduce the risk index RF0(t) for the free boundary problem, which depends on the advection coefficient and time. Sufficient conditions for the disease to prevail or not are obtained. Our results suggest that the disease must spread if RF0(t0) ≤ 1 for some t0 and the disease is vanishing if RF0(∞) < 1, while if RF0 (0) < 1, the spreading or vanishing of the disease depends on the initial state of infected individuals as well as the expanding capability of the free boundary. We also illustrate the impacts of the expanding capability on the spreading fronts via the numerical simulations.
本文提出并分析了一种用于传染病传播的反应扩散对流SIR模型。引入自由边界来描述疾病的传播前沿。通过给出具有狄利克雷边界条件的相关模型的基本再生数(R_{DA0})，我们引入了自由边界问题的风险指数(R_{F0}(t))，它取决于对流系数和时间。得到了疾病流行与否的充分条件。我们的结果表明，如果对于某个(t_0)有(R_{F0}(t_0) \leq 1)，则疾病必然传播；如果(R_{F0}(\infty) < 1)，则疾病消失；而如果(R_{F0}(0) < 1)，疾病的传播或消失取决于感染个体的初始状态以及自由边界的扩展能力。我们还通过数值模拟说明了扩展能力对传播前沿的影响。
