# 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 and Continuous Dynamical Systems - B, 2017, 22 (6) : 2261-2290. doi: 10.3934/dcdsb.2017095
##### References:
 [1] J. R. Andrews and S. Basu, Transmission dynamics and control of cholera in Haiti: An epidemic model, Lancet, 377 (2011), 1248-1255. [2] E. Bertuzzo, L. Mari, L. Righetto, M. Gatto, R. Casagrandi, M. Blokesch, I. Rodriguez-Iturbe and A. Rinaldo, Prediction of the spatial evolution and effects of control measures for the unfolding Haiti cholera outbreak, Geophys. Res. Lett., 38 (2011), L06403. [3] D. L. Chao, M. E. Halloran and I. M. Longini, Vaccination strategies for epidemic cholera in Haiti with implications for the developing world, Proc. Natl. Acad. Sci., 108 (2011), 7081-7085. [4] C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infect. Dis., 1 (2001), p1. [5] M. C. Eisenberga, G. Kujbidad, A. R. Tuited, D. N. Fismand and J. H. Tiena, Examining rainfall and cholera dynamics in Haiti using statistical and dynamic modeling approaches, Epidemics, 5 (2013), 197-207. [6] S. M. Faruque, I. B. Naser, M. J. Islam, A. S. G. Faruque, A. N. Ghosh, G. B. Nair, D. A. Sack and J. J. Mekalanos, Seasonal epidemics of cholera inversely correlate with the prevalence of environmental cholera phages, Proc. Nat. Acad. Sci., 102 (2004), 1702-1707. [7] J. K. Hale, Ordinary Differential Equations, Pure and Applied Mathematics, Vol. XXI, WileyInterscience, New York, 1969. [8] D. M. Hartley, J. G. Morris and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLoS Medicine, 3 (2006), e7. [9] M. A. Jensen, S. M. Faruque, J. J. Mekalanos and B. R. levin, Modeling the role of bacteriophage in the control of cholera outbreaks, PNAS, 103 (2006), 4652-4657. [10] R. I. Joh, H. Wang, H. Weiss and J. S. Weitz, Dynamics of indirectly transmitted infectious diseases with immunological threshold, Bull. Math. Bio., 71 (2009), 845-862. [11] J. D. Kong, W. Davis, X. Li and H. Wang, Stability and sensitivity analysis of the iSIR model for indirectly transmitted infectious dieases with immunological threshold, SIAM J. Appl. Math., 74 (2014), 1418-1441. [12] S. Liao and J. Wang, Stability analysis and application of a mathematical cholera model, Math. Biosci. and Eng., 8 (2011), 733-752. [13] Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D. L. Smith and J. G. Morris Jr., Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proc. Nat. Acad. Sci., 108 (2011), 8767-8772. [14] E. J. Nelson, J. B. Harris, J. G. Morris, S. B. Calderwood and A. Camilli, Cholera transmission: The host, pathogen and bacteriophage dynamics, Nature Reviews: Microbiology, 7 (2009), 693-702. [15] L. Righetto, E. Bertuzzo, L. Mari, E. Schild, R. Casagrandi, M. Gatto, I. Rodriguez-Iturbe and A. Rinaldo, Rainfall mediations in the spreading of epidemic cholera, Advances in Water Resources, 60 (2013), 34-46. [16] L. Righetto, R. Casagrandi, E. Bertuzzo, L. Mari, M. Gatto, I. Rodriguez-Iturbe and A. Rinaldo, The role of aquatic reservoir fluctuations in long-term cholera patterns, Epidemics, 4 (2012), 33-42. [17] F. L. Thompson, B. Austin and J. Swings, The Biology of Vibrios, ASM Press, Washington, D.C., 2006. [18] J. P. Tian and J. Wang, Global stability for cholera epidemic models, Math. Biosci., 232 (2011), 31-41. [19] J. H. Tien and D. J. D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull. Math. Bio., 72 (2010), 1502-1533. [20] A. L. Tuite, J. Tien, M. Eisenberg, D. J. D. Earn, J. Ma and D. N. Fisman, Cholera epidemic in Haiti, 2010: Using a transmission model to explain spatial spread of disease and identify optimal control interventions, Ann. Intern. Med., 154 (2011), 593-601. [21] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition, Springer, Berlin, 1996. [22] J. Wang and S. Liao, A generalized cholera model and epidemic-endemic analysis, J. Biol. Dyn., 6 (2012), 568-589. [23] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutins, Appl. Math. Science, Vol. 14, Springer-Verlag, 1975. [24] World Health Organization (WHO) web page: http://www.who.org.

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##### References:
 [1] J. R. Andrews and S. Basu, Transmission dynamics and control of cholera in Haiti: An epidemic model, Lancet, 377 (2011), 1248-1255. [2] E. Bertuzzo, L. Mari, L. Righetto, M. Gatto, R. Casagrandi, M. Blokesch, I. Rodriguez-Iturbe and A. Rinaldo, Prediction of the spatial evolution and effects of control measures for the unfolding Haiti cholera outbreak, Geophys. Res. Lett., 38 (2011), L06403. [3] D. L. Chao, M. E. Halloran and I. M. Longini, Vaccination strategies for epidemic cholera in Haiti with implications for the developing world, Proc. Natl. Acad. Sci., 108 (2011), 7081-7085. [4] C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infect. Dis., 1 (2001), p1. [5] M. C. Eisenberga, G. Kujbidad, A. R. Tuited, D. N. Fismand and J. H. Tiena, Examining rainfall and cholera dynamics in Haiti using statistical and dynamic modeling approaches, Epidemics, 5 (2013), 197-207. [6] S. M. Faruque, I. B. Naser, M. J. Islam, A. S. G. Faruque, A. N. Ghosh, G. B. Nair, D. A. Sack and J. J. Mekalanos, Seasonal epidemics of cholera inversely correlate with the prevalence of environmental cholera phages, Proc. Nat. Acad. Sci., 102 (2004), 1702-1707. [7] J. K. Hale, Ordinary Differential Equations, Pure and Applied Mathematics, Vol. XXI, WileyInterscience, New York, 1969. [8] D. M. Hartley, J. G. Morris and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLoS Medicine, 3 (2006), e7. [9] M. A. Jensen, S. M. Faruque, J. J. Mekalanos and B. R. levin, Modeling the role of bacteriophage in the control of cholera outbreaks, PNAS, 103 (2006), 4652-4657. [10] R. I. Joh, H. Wang, H. Weiss and J. S. Weitz, Dynamics of indirectly transmitted infectious diseases with immunological threshold, Bull. Math. Bio., 71 (2009), 845-862. [11] J. D. Kong, W. Davis, X. Li and H. Wang, Stability and sensitivity analysis of the iSIR model for indirectly transmitted infectious dieases with immunological threshold, SIAM J. Appl. Math., 74 (2014), 1418-1441. [12] S. Liao and J. Wang, Stability analysis and application of a mathematical cholera model, Math. Biosci. and Eng., 8 (2011), 733-752. [13] Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D. L. Smith and J. G. Morris Jr., Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proc. Nat. Acad. Sci., 108 (2011), 8767-8772. [14] E. J. Nelson, J. B. Harris, J. G. Morris, S. B. Calderwood and A. Camilli, Cholera transmission: The host, pathogen and bacteriophage dynamics, Nature Reviews: Microbiology, 7 (2009), 693-702. [15] L. Righetto, E. Bertuzzo, L. Mari, E. Schild, R. Casagrandi, M. Gatto, I. Rodriguez-Iturbe and A. Rinaldo, Rainfall mediations in the spreading of epidemic cholera, Advances in Water Resources, 60 (2013), 34-46. [16] L. Righetto, R. Casagrandi, E. Bertuzzo, L. Mari, M. Gatto, I. Rodriguez-Iturbe and A. Rinaldo, The role of aquatic reservoir fluctuations in long-term cholera patterns, Epidemics, 4 (2012), 33-42. [17] F. L. Thompson, B. Austin and J. Swings, The Biology of Vibrios, ASM Press, Washington, D.C., 2006. [18] J. P. Tian and J. Wang, Global stability for cholera epidemic models, Math. Biosci., 232 (2011), 31-41. [19] J. H. Tien and D. J. D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull. Math. Bio., 72 (2010), 1502-1533. [20] A. L. Tuite, J. Tien, M. Eisenberg, D. J. D. Earn, J. Ma and D. N. Fisman, Cholera epidemic in Haiti, 2010: Using a transmission model to explain spatial spread of disease and identify optimal control interventions, Ann. Intern. Med., 154 (2011), 593-601. [21] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition, Springer, Berlin, 1996. [22] J. Wang and S. Liao, A generalized cholera model and epidemic-endemic analysis, J. Biol. Dyn., 6 (2012), 568-589. [23] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutins, Appl. Math. Science, Vol. 14, Springer-Verlag, 1975. [24] World Health Organization (WHO) web page: http://www.who.org.
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 $^{-1}$
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