Seasonal forcing and exponential threshold incidence in cholera dynamics

Jinhuo Luo; Jin Wang; Hao Wang

doi:10.3934/dcdsb.2017095

Article Contents

2017, Volume 22, Issue 6: 2261-2290. Doi: 10.3934/dcdsb.2017095

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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

Author Bio: E-mail address: jhluo@shou.edu.cn; E-mail address: Jin-Wang02@utc.edu
The third author's research was partially supported by NSERC. E-mail address: hao8@ualberta.ca
The third author's research was partially supported by NSERC. E-mail address: hao8@ualberta.ca

Received: February 28, 2015

Revised: December 31, 2016

Published: May 2017

The second author's research was partially supported by NSF.

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Abstract

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.

Keywords:

Mathematics Subject Classification: 93A30, 37B55, 34D20, 34D23, 97M10, 34C25, 37D35, 34C60.

Citation:

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Figure 1. Satuation function $\alpha(\cdot)$ of Holling-type

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Figure 2. Saturation function $\alpha(\cdot)$ of exponential type

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Figure 3. System (40) possesses a bistability (See column 1, column 2)

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Figure 4. An example when system (40) does not approach $(1, 0, K)$ in the case that $c$ is slightly greater than $K$

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Figure 5. Populations of system (5) approach a periodic solution

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Figure 6. 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

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Figure 7. Two final states of system (5) depending on different initial values

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Figure 8. The third final state of system (5) and the locally enlarged figure (shown in the right column)

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Figure 9. Periodic outbreak of epidemic ($\xi=0$, left column), and durative infection ($\xi=90$, right column)

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Figure 10. System encounters a sudden event. Left column: $N=1\times 10^{6}$. Right column: $N=1\times 10^{7}$

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Figure 11. Curves of $u(B)$ and $v(B)$ have an unique intersection $\bar{B}$

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Figure 12. Curves of function $f$ and $g$ with changing threshold values $c$

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Figure 13. System (40) has two equilibria

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Figure 14. System (40) has three equilibria

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Figure 15. System (40) has four equilibria

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Table 1. 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}$

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