Aili Abulajiang, Teng Zhidong, Zhang Long
College of Mathematics and System Science, Xinjiang University, Urumqi 830046, Xinjiang, China.
College of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830017, Xinjiang, China.
Math Biosci Eng. 2023 Jan;20(1):505-533. doi: 10.3934/mbe.2023023. Epub 2022 Oct 11.
In this paper, a coupling SEIR epidemic model is proposed to characterize the interaction of virus spread in the body of hosts and between hosts with environmentally-driven infection, humoral immunity and incubation of disease. The threshold criteria on the local (or global) stability of feasible equilibria with or without antibody response are established. The basic reproduction number $ R_{b0} $ is obtained for the SEIR model without an antibody response, by which we find that the disease-free equilibrium is locally asymptotically stable if $ R_{b0} < 1 $. Two endemic equilibria exist if $ R_{b0} < 1 $, in which one is locally asymptotically stable under some additional conditions but the other is unstable, which means there is backward bifurcation. In addition, the uniform persistence of this model is discussed. For the SEIR model with an antibody response, the basic reproduction number $ R_{0} $ is calculated, from which the disease-free equilibrium is globally asymptotically stable if $ R_0\leq1 $, and the unique endemic equilibrium is globally asymptotically stable if $ R_0 > 1 $. Antibody immunity in the host plays a great role in the control of disease transmission, especially when the diseases between the hosts are entirely extinct once antibody cells in the host reach a proper level. Finally, the main conclusions are illustrated by some special examples and numerical simulations.
本文提出了一个耦合的SEIR传染病模型，以刻画病毒在宿主体内传播以及在宿主间传播过程中，受环境驱动的感染、体液免疫和疾病潜伏期之间的相互作用。建立了有或无抗体反应时可行平衡点的局部（或全局）稳定性的阈值标准。得到了无抗体反应的SEIR模型的基本再生数$R_{b0}$，由此发现当$R_{b0}<1$时，无病平衡点是局部渐近稳定的。当$R_{b0}>1$时存在两个地方病平衡点，其中一个在一些附加条件下是局部渐近稳定的，而另一个是不稳定的，这意味着存在向后分支。此外，还讨论了该模型的一致持久性。对于有抗体反应的SEIR模型，计算了基本再生数$R_0$，由此可知当$R_0\leq1$时无病平衡点是全局渐近稳定的，当$R_0>1$时唯一的地方病平衡点是全局渐近稳定的。宿主中的抗体免疫在疾病传播控制中起着重要作用，特别是当宿主中的抗体细胞达到适当水平时，宿主间的疾病会完全灭绝。最后，通过一些特殊例子和数值模拟说明了主要结论。
