Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, Yaoundé P.O. Box 812, Cameroon.
Department of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, Tallinn 10120, Estonia; Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes 20131, Mexico.
Comput Methods Programs Biomed. 2021 Nov;212:106469. doi: 10.1016/j.cmpb.2021.106469. Epub 2021 Oct 19.
In this work, we analyze the spatial-temporal dynamics of a susceptible-infected-recovered (SIR) epidemic model with time delays. To better describe the dynamical behavior of the model, we take into account the cumulative effects of diffusion in the population dynamics, and the time delays in both the Holling type II treatment and the disease transmission process, respectively.
We perform linear stability analyses on the disease-free and endemic equilibria. We provide the expression of the basic reproduction number and set conditions on the backward bifurcation using Castillo's theorem. The values of the critical time transmission, the treatment delays and the relationship between them are established.
We show that the treatment rate decreases the basic reproduction number while the transmission rate significantly affects the bifurcation process in the system. The transmission and treatment time-delays are found to be inversely proportional to the susceptible and infected diffusion rates. The analytical results are numerically tested. The results show that the treatment rate significantly reduces the density of infected population and ensures the transition from the unstable to the stable domain. Moreover, the system is more sensible to the treatment in the stable domain.
The density of infected population increases with respect to the infected and susceptible diffusion rates. Both effects of treatment and transmission delays significantly affect the behavior of the system. The transmission time-delay at the critical point ensures the transition from the stable (low density) to the unstable (high density) domain.
在这项工作中，我们分析了具有时滞的易感染-感染-恢复（SIR）传染病模型的时空动力学。为了更好地描述模型的动力学行为，我们考虑了人口动力学中扩散的累积效应，以及分别在霍林型 II 治疗和疾病传播过程中的时滞。
我们对无病平衡点和地方病平衡点进行线性稳定性分析。我们使用卡斯蒂略定理给出了基本再生数的表达式，并给出了后分叉的条件。建立了临界传输时间、治疗延迟时间和它们之间关系的表达式。
我们表明，治疗率降低了基本再生数，而传播率显著影响了系统中的分岔过程。发现传输和治疗时滞与易感染和感染扩散率成反比。分析结果通过数值测试进行了验证。结果表明，治疗率显著降低了感染人群的密度，并确保了从不稳定到稳定域的转变。此外，系统在稳定域对治疗更为敏感。
感染人群的密度随感染和易感染扩散率的增加而增加。治疗和传输延迟的双重影响显著影响了系统的行为。临界点的传输时滞确保了从稳定（低密度）到不稳定（高密度）域的转变。
