Article Contents
Article Contents

# Seasonal forcing and exponential threshold incidence in cholera dynamics

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

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

 Citation:

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

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

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

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

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

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

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

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

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

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

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

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

Figure 13.  System (40) has two equilibria

Figure 14.  System (40) has three equilibria

Figure 15.  System (40) has four equilibria

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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