School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, Hubei, People's Republic of China.
Present Address: Department of Mathematics, Cleveland State University, Cleveland, 44115, Ohio, USA.
J Math Biol. 2023 Mar 6;86(4):52. doi: 10.1007/s00285-023-01887-8.
In this paper, we propose a two-patch SIRS model with a nonlinear incidence rate: [Formula: see text] and nonconstant dispersal rates, where the dispersal rates of susceptible and recovered individuals depend on the relative disease prevalence in two patches. In an isolated environment, the model admits Bogdanov-Takens bifurcation of codimension 3 (cusp case) and Hopf bifurcation of codimension up to 2 as the parameters vary, and exhibits rich dynamics such as multiple coexistent steady states and periodic orbits, homoclinic orbits and multitype bistability. The long-term dynamics can be classified in terms of the infection rates [Formula: see text] (due to single contact) and [Formula: see text] (due to double exposures). In a connected environment, we establish a threshold [Formula: see text] between disease extinction and uniform persistence under certain conditions. We numerically explore the effect of population dispersal on disease spread when [Formula: see text] and patch 1 has a lower infection rate, our results indicate: (i) [Formula: see text] can be nonmonotonic in dispersal rates and [Formula: see text] ([Formula: see text] is the basic reproduction number of patch i) may fail; (ii) the constant dispersal of susceptible individuals (or infective individuals) between two patches (or from patch 2 to patch 1) will increase (or reduce) the overall disease prevalence; (iii) the relative prevalence-based dispersal may reduce the overall disease prevalence. When [Formula: see text] and the disease outbreaks periodically in each isolated patch, we find that: (a) small unidirectional and constant dispersal can lead to complex periodic patterns like relaxation oscillations or mixed-mode oscillations, whereas large ones can make the disease go extinct in one patch and persist in the form of a positive steady state or a periodic solution in the other patch; (b) relative prevalence-based and unidirectional dispersal can make periodic outbreak earlier.
本文提出了一个具有非线性感染率和非常数扩散率的两斑块 SIRS 模型：[公式：见正文]，其中易感染者和恢复者的扩散率取决于两个斑块中相对疾病流行率。在隔离环境中，当参数变化时，模型存在余维 3 的 Bogdanov-Takens 分歧（尖点情况）和余维至多 2 的 Hopf 分歧，并表现出丰富的动力学，如多个共存的稳定状态和周期轨道、同宿轨道和多型双稳性。长期动力学可以根据感染率[公式：见正文]（由于单次接触）和[公式：见正文]（由于双重暴露）来分类。在连通环境中，我们在一定条件下建立了疾病灭绝和均匀持续的阈值[公式：见正文]。当[公式：见正文]和斑块 1 具有较低的感染率时，我们数值研究了种群扩散对疾病传播的影响，结果表明：（i）[公式：见正文]在扩散率中可能是非单调的，[公式：见正文]（[公式：见正文]是斑块 i 的基本再生数）可能失效；（ii）两个斑块之间易感染者（或感染者）的常数扩散（或从斑块 2 到斑块 1）将增加（或减少）总疾病流行率；（iii）基于相对流行率的扩散可能会降低总疾病流行率。当[公式：见正文]且疾病在每个孤立斑块中周期性爆发时，我们发现：（a）小的单向和常数扩散可能导致复杂的周期模式，如松弛振荡或混合模式振荡，而大的扩散可能导致疾病在一个斑块中灭绝，并以另一个斑块中的正稳态或周期解的形式持续存在；（b）基于相对流行率的单向扩散可以使周期性爆发更早发生。
