# American Institute of Mathematical Sciences

August  2017, 22(6): 2261-2290. doi: 10.3934/dcdsb.2017095

## Seasonal forcing and exponential threshold incidence in cholera dynamics

 1 College of Information Technology, Shanghai Ocean University, Shanghai 201306, China 2 Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, United States 3 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada

The third author's research was partially supported by NSERC. E-mail address: hao8@ualberta.ca

Received  March 2015 Revised  January 2017 Published  March 2017

Fund Project: The second author's research was partially supported by NSF

We propose a seasonal forcing iSIR (indirectly transmitted SIR) model with a modified incidence function, due to the fact that the seasonal fluctuations can be the main culprit for cholera outbreaks. For this nonautonomous system, we provide a sufficient condition for the persistence and the existence of a periodic solution. Furthermore, we provide a sufficient condition for the global stability of the periodic solution. Finally, we present some simulation examples for both autonomous and nonautonomous systems. Simulation results exhibit dynamical complexities, including the bistability of the autonomous system, an unexpected outbreak of cholera for the nonautonomous system, and possible outcomes induced by sudden weather events. Comparatively the nonautonomous system is more realistic in describing the indirect transmission of cholera. Our study reveals that the relative difference between the value of immunological threshold and the peak value of bacterial biomass is critical in determining the dynamical behaviors of the system.

Citation: Jinhuo Luo, Jin Wang, Hao Wang. Seasonal forcing and exponential threshold incidence in cholera dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2261-2290. doi: 10.3934/dcdsb.2017095
##### References:

show all references

##### References:
Satuation function $\alpha(\cdot)$ of Holling-type
Saturation function $\alpha(\cdot)$ of exponential type
System (40) possesses a bistability (See column 1, column 2)
An example when system (40) does not approach $(1, 0, K)$ in the case that $c$ is slightly greater than $K$
Populations of system (5) approach a periodic solution
When the threshold of immunity is significantly higher than the maximum of bacterial capacity, populations of system (5) tend to a disease free periodic solution
Two final states of system (5) depending on different initial values
The third final state of system (5) and the locally enlarged figure (shown in the right column)
Periodic outbreak of epidemic ($\xi=0$, left column), and durative infection ($\xi=90$, right column)
System encounters a sudden event. Left column: $N=1\times 10^{6}$. Right column: $N=1\times 10^{7}$
Curves of $u(B)$ and $v(B)$ have an unique intersection $\bar{B}$
Curves of function $f$ and $g$ with changing threshold values $c$
System (40) has two equilibria
System (40) has three equilibria
System (40) has four equilibria
Parameter values from Jensen et al. [9]
 Parameter Values Description Units $r$ 0.2-14.3 Maximum per capita pathogen growth rate day $^{-1}$ $K$ $10^6$ Pathogen carrying capacity cell liter $^{-1}$ $H$ $10^6-10^8$ Half-saturation pathogen density cell liter $^{-1}$ $a$ 0.08 -0.12 Maximum rate of infection day $^{-1}$ $\delta$ 0.1 Recovery rate day $^{-1}$ $\xi$ 10-100 Pathengen shed rate cell liter $^{-1}\text{day} ^{-1}$ $\mu$ $5\times 10^{-5}-5\times 10^{-4}$ Natural human birth/death rate day $^{-1}$ $N$ $10^6$ Total Population persons $c$ $\approx 10^6$ Minimum infection dose cell liter $^{-1}$
 Parameter Values Description Units $r$ 0.2-14.3 Maximum per capita pathogen growth rate day $^{-1}$ $K$ $10^6$ Pathogen carrying capacity cell liter $^{-1}$ $H$ $10^6-10^8$ Half-saturation pathogen density cell liter $^{-1}$ $a$ 0.08 -0.12 Maximum rate of infection day $^{-1}$ $\delta$ 0.1 Recovery rate day $^{-1}$ $\xi$ 10-100 Pathengen shed rate cell liter $^{-1}\text{day} ^{-1}$ $\mu$ $5\times 10^{-5}-5\times 10^{-4}$ Natural human birth/death rate day $^{-1}$ $N$ $10^6$ Total Population persons $c$ $\approx 10^6$ Minimum infection dose cell liter $^{-1}$
 [1] Suqi Ma. Low viral persistence of an immunological model. Mathematical Biosciences & Engineering, 2012, 9 (4) : 809-817. doi: 10.3934/mbe.2012.9.809 [2] Kazuo Yamazaki, Xueying Wang. Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 559-579. doi: 10.3934/mbe.2017033 [3] Kaifa Wang, Aijun Fan. Uniform persistence and periodic solution of chemostat-type model with antibiotic. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 789-795. doi: 10.3934/dcdsb.2004.4.789 [4] M. P. Moschen, A. Pugliese. The threshold for persistence of parasites with multiple infections. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1483-1496. doi: 10.3934/cpaa.2008.7.1483 [5] Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169 [6] Zhijun Liu, Weidong Wang. Persistence and periodic solutions of a nonautonomous predator-prey diffusion with Holling III functional response and continuous delay. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 653-662. doi: 10.3934/dcdsb.2004.4.653 [7] Zhenguo Bai, Yicang Zhou. Threshold dynamics of a bacillary dysentery model with seasonal fluctuation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 1-14. doi: 10.3934/dcdsb.2011.15.1 [8] Antoine Perasso. Global stability and uniform persistence for an infection load-structured SI model with exponential growth velocity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 15-32. doi: 10.3934/cpaa.2019002 [9] Xueping Li, Jingli Ren, Sue Ann Campbell, Gail S. K. Wolkowicz, Huaiping Zhu. How seasonal forcing influences the complexity of a predator-prey system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 785-807. doi: 10.3934/dcdsb.2018043 [10] Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641 [11] Shu Liao, Jin Wang. Stability analysis and application of a mathematical cholera model. Mathematical Biosciences & Engineering, 2011, 8 (3) : 733-752. doi: 10.3934/mbe.2011.8.733 [12] Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365 [13] Wen Jin, Horst R. Thieme. An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 447-470. doi: 10.3934/dcdsb.2016.21.447 [14] Xuewei Ju, Desheng Li. Global synchronising behavior of evolution equations with exponentially growing nonautonomous forcing. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1921-1944. doi: 10.3934/cpaa.2018091 [15] Pierre Gabriel. Global stability for the prion equation with general incidence. Mathematical Biosciences & Engineering, 2015, 12 (4) : 789-801. doi: 10.3934/mbe.2015.12.789 [16] Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631 [17] Guihong Fan, Yijun Lou, Horst R. Thieme, Jianhong Wu. Stability and persistence in ODE models for populations with many stages. Mathematical Biosciences & Engineering, 2015, 12 (4) : 661-686. doi: 10.3934/mbe.2015.12.661 [18] Alan E. Lindsay, Michael J. Ward. An asymptotic analysis of the persistence threshold for the diffusive logistic model in spatial environments with localized patches. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1139-1179. doi: 10.3934/dcdsb.2010.14.1139 [19] Zhengxin Zhou. On the Poincaré mapping and periodic solutions of nonautonomous differential systems. Communications on Pure & Applied Analysis, 2007, 6 (2) : 541-547. doi: 10.3934/cpaa.2007.6.541 [20] Yoichi Enatsu, Yukihiko Nakata. Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate. Mathematical Biosciences & Engineering, 2014, 11 (4) : 785-805. doi: 10.3934/mbe.2014.11.785

2018 Impact Factor: 1.008