一类具有预防接种的两菌株共感模型的传染病动力学分析

doi:10.6040/j.issn.1671-9352.0.2022.543

摘要/Abstract

摘要：

为了探讨多菌株在同一宿主群体有共同感染的传播动态，建立并分析了持续接种一种菌株疫苗后两菌株共同传播的动力学数学模型。首先通过对模型的计算和分析，得到4类平衡点存在的充分条件，除了无病平衡点和2个单株地方病平衡点以外，模型还存在菌株1、2都共存的地方病平衡点；其次，利用Lyapunov稳定性定理证明当2个菌株的基本再生数都小于1时，无病平衡点是全局稳定的。在确定单菌株地方病平衡点的稳定性时，引入了入侵再生数，当对应入侵再生数小于1时该菌株的地方病平衡点是局部稳定的；然后利用Castillo-Chavez和Song分支定理，证明了该模型不存在后向分支现象，进而证明了2个菌株的基本再生数都大于1时，共存平衡点是局部渐近稳定的；最后，通过数值模拟验证了以上的结论。

关键词: 多菌株, 共同感染, 基本再生数, 全局稳定性, 地方病平衡点, 入侵再生数

Abstract:

To explore the dynamics of co-infection of multiple strains in the same host population, a mathematical model of co-transmission dynamics of two strains after continuous inoculation with strain 1 vaccine is established and analyzed. Firstly, the sufficient conditions for the existence of four equilibrium points are obtained by calculating and analyzing the model. In addition to the disease-free equilibrium point and the two single endemic equilibrium points, the model also has an endemic equilibrium point where both strains 1 and 2 coexist. Secondly, Lyapunov stability theorem is used to prove that the disease-free equilibrium is globally stable when the basic reproduction number of two strains is less than 1. The invasion-reproduction number is introduced to determine the stability of the single-strain endemic equilibrium point. When the corresponding invasion reproduction number is less than 1, the endemic equilibrium point of the strain is locally stable. Then, using Castillo-Chavez and Song?s bifurcation theorem, it is proved that the model does not have backward bifurcation phenomenon, and then it is proved that the coexistence equilibrium point is locally asymptotically stable when the basic reproduction number of the two strains is greater than 1. Finally, the above conclusions are verified by numerical simulation.

Key words: multiple strains, co-infection, basic reproduction number, global stability, endemic equilibrium, invasion reproduction number

中图分类号:

O175

陈刚,张睿. 一类具有预防接种的两菌株共感模型的传染病动力学分析[J]. 《山东大学学报(理学版)》, 2023, 58(10): 84-96.

Gang CHEN,Rui ZHANG. Dynamics of a two-strain co-infection epidemic model with vaccination[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(10): 84-96.

图/表 4

图1

表1

表2

图2

参考文献 21

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参数	参数描述	参数	参数描述
Λ	外来人口输入率	γ₃	共感者同时对菌株1和菌株2的恢复率
μ	区域内人口自然死亡率	δ₁β₁	共感者对菌株2感染者的传播率
ρ	输入人口菌株1疫苗的接种率(ρ < 1)	δ₂β₂	共感者对菌株1感染者的传播率
ν	单位时间菌株1疫苗持续的接种率	α₁、α₂	菌株1、2感染者的恢复率
β₁、β₂	菌株1/2感染者对易感人群的有效接触率	d₁、d₂	菌株1、2感染者的因病致死率
β₃	共感者同时对易感人群的菌株1，2的传播率	d₃	共感者的因病致死率
γ₁、γ₂	共感者对菌株1、2的恢复率	σ	恢复人群对免疫能力的消失率

参数	图 2(a)取值	图 2(b)取值	图 2(c)取值	图 2(d)取值
Λ	0.070	0.070	0.070	0.070
μ	3.5×10^-5	3.5×10^-5	3.5×10^-5	3.5×10^-5
ρ	5.0×10^-3	5.0×10^-3	5.0×10^-3	5.0×10^-3
ν	1.0×10^-5	1.0×10^-5	1.0×10^-5	1.0×10^-5
β₁	0.050	0.040	0.060	0.013
β₂	0.050	0.080	0.030	0.075
β₃	0.050	0.015	0.020	0.050
γ₁	0.045	0.030	0.030	0.030
γ₂	0.030	0.025	0.025	0.038
γ₃	0.020	0.050	0.050	0.050
δ₁	0.800	0.800	1.100	0.800
δ₂	0.600	0.400	1.300	0.400
α₁	0.060	0.018	0.020	0.020
α₂	0.060	0.025	0.035	0.030
d₁	5.5×10^-6	5.5×10^-6	5.5×10^-6	5.5×10^-6
d₂	5.5×10^-6	5.5×10^-6	5.5×10^-6	5.5×10^-6
d₃	9.0×10^-6	9.0×10^-6	9.0×10^-6	9.0×10^-6
σ	0.080	0.080	0.080	0.080

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