Dynamics of a Cholera model with environmental transmission and physiological age structure

Abstract

Abstract: Based on the rules of Cholera transmission between humans and environment, as well as individual differences between humans, a Cholera model with environmental transmission and physiological age structure is proposed, and the existence and uniqueness of the global positive solution of this model are proved by semigroup theory. Furthermore, the expression of basic reproduction number R₀ is derived by linear approximation method, it is concluded that if R₀<1, the disease-free steady state is globally asymptotically stable; if R₀>1, the disease-free steady state is unstable, and the model has a unique endemic steady state and is locally asymptotically stable under certain conditions. The main theoretical results are explained by numerical simulation.

Key words: cholera, age-structured, environmental transmission, basic reproduction number, steady state and stability

CLC Number:

O175

KANG Yue-nan, GUO Wen-dan, NIE Lin-fei. Dynamics of a Cholera model with environmental transmission and physiological age structure[J]. Journal of Lanzhou University of Technology, 2023, 49(2): 157-164.

References

[1] 世界卫生组织.霍乱 [EB/OL].[2022-02-14].https://www.who.int/zh/news-room/fact-sheets/detail/cholera.
[2] BAI N,SONG C W,XU R.Mathematical analysis and application of a cholera transmission model with waning vaccine-induced immunity [J].Nonlinear Analysis:Real World Applications,2021,58:103232.
[3] 廖书,杨炜明.一类非标准离散霍乱动力学模型 [J].工程数学学报,2019,36(1):85-98.
[4] SHARMA S,SINGH F.Bifurcation and stability analysis of a cholera model with vaccination and saturated treatment [J].Chaos,Solitons and Fractals,2021,146:110912.
[5] TIAN X H,XU R,LIN J Z.Mathematical analysis of a cholera infection model with vaccination strategy [J].Applied Mathematics and Computation,2019,361:517-535.
[6] WANG S F,HU L,NIE L F.Global dynamics and optimal control of an age-structure Malaria transmission model with vaccination and relapse [J].Chaos,Solitons and Fractals,2021,150:111216.
[7] 孙丹丹,李盈科,滕志东,等.具有年龄结构的麻疹传染病模型的稳定性分析 [J].数学物理学报,2021,41(6):1950-1968.
[8] LIN J Z,XU R,TIAN X H.Global dynamics of an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence [J].Applied Mathematical Modelling,2018,63:688-708.
[9] YANG J Y,MODNAK C,WANG J.Dynamical analysis and optimal control simulation for an age-structured cholera transmission model [J].Journal of the Franklin Institute-Engineering and Applied Mathematics,2019,356(15):8438-8467.
[10] 梁霜霜,聂麟飞,胡琳.具有年龄结构和水平传播的媒介传染病模型研究 [J].华东师范大学学报(自然科学版),2021(3):47-55.
[11] CHA Y,IANNELLI M,MILNER F A.Existence and uniqueness of endemic states for the age-structured SIR epidemic model [J].Mathematical Biosciences,1998,150(2):117-133.
[12] WEBB G F.Theory of nonlinear age-dependent population dynamics [M].New York:Marcel Dekker,1985.
[13] INABA H.Threshold and stability results for an age-structured epidemic model [J].Journal of Mathematical Biology,1990,28:411-434.
[14] DIEKMANN O,HEESTERBEEK J A P,METZ J A J.On the definition and the computation of the basic reproduction ratio R₀ in models for infectious diseases in heterogeneous populations [J].Journal of Mathematical Biology,1990,28(4):365-382.
[15] SCHMEIDLER D.Fatou's lemma in several dimensions [J].Proceedings of the American Mathematical Society,1970,24(2):300-306.