A mathematical model for the seasonal transmission of schistosomiasis in the lake and marshland regions of China

Yingke Li; Zhidong Teng; Shigui Ruan; Mingtao Li; Xiaomei Feng

doi:10.3934/mbe.2017066

Article Contents

2017, Volume 14, Issue 5&6: 1279-1299. Doi: 10.3934/mbe.2017066

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A mathematical model for the seasonal transmission of schistosomiasis in the lake and marshland regions of China

a.
College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, China
b.
College of Mathematics and Physics, Xinjiang Agriculture University, Urumqi, Xinjiang 830052, China
c.
Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA
d.
Complex Systems Research Center, Shanxi University, Taiyuan, Shanxi 030006, China
e.
Department of Mathematics, Yuncheng University, Yuncheng, Shanxi 044000, China

^* Corresponding authorr: Zhidong Teng (E-mail: zhidong1960@163.com)
^* Corresponding authorr: Zhidong Teng (E-mail: zhidong1960@163.com)

Received: April 04, 2016

Revised: October 29, 2016

Published: November 2017

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Abstract

Schistosomiasis, a parasitic disease caused by Schistosoma Japonicum, is still one of the most serious parasitic diseases in China and remains endemic in seven provinces, including Hubei, Anhui, Hunan, Jiangsu, Jiangxi, Sichuan, and Yunnan. The monthly data of human schistosomiasis cases in Hubei, Hunan, and Anhui provinces (lake and marshland regions) released by the Chinese Center for Disease Control and Prevention (China CDC) display a periodic pattern with more cases in late summer and early autumn. Based on this observation, we construct a deterministic model with periodic transmission rates to study the seasonal transmission dynamics of schistosomiasis in these lake and marshland regions in China. We calculate the basic reproduction number $R_{0}$, discuss the dynamical behavior of solutions to the model, and use the model to fit the monthly data of human schistosomiasis cases in Hubei. We also perform some sensitivity analysis of the basic reproduction number $R_{0}$ in terms of model parameters. Our results indicate that treatment of at-risk population groups, improving sanitation, hygiene education, and snail control are effective measures in controlling human schistosomiasis in these lakes and marshland regions.

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Mathematics Subject Classification: Primary: 37N25, 93D30; Secondary: 92B05.

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Figure 1. The human cases in Hunan, Anhui and Hubei from January 2008 to December 2011

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Figure 2. Simplified life cycle of human schistosomiasis

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Figure 3. Transmission diagram of schistosomiasis among human, snail, and miracidia and cercariae in water

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Figure 4. The $12$-periodic solutions in Example 5.3 when $R_{0}=1.8048>1.$

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Figure 5. Comparison between the reported human schistosomiasis cases in Hubei from January 2008 to December 2014 and the simulation of $I_{H}(t)$ from model (1)

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Figure 6. Disease development trend by forecasting model (1). The parameter and initial values are the same as in Figure 5

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Figure 7. Tendency of human schistosomiasis cases with different $R_{0}$: (a) $\gamma_{H}=0.131$, $R_{0}=1.2387$; (b) $\gamma_{H}=0.207$, $R_{0}=0.9971$. All other parameter values are the same as in Table 1

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Figure 8. The influence of parameters on $R_{0}$: (a) versus $\Lambda_{V}$, (b) versus $\lambda_{M}$, (c) versus $\lambda_{P}$, (d) versus $\gamma_{H}$. Other parameter values are unchanged as in Table 1

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Figure 9. The influence of different values on $I_{H}(t)$: (a) different values of $\Lambda_V$, (b) different values of $\mu_V$, (c) different values of $\lambda_M$, (d) different values of $\lambda_P$, (e) different values of $a_H$, (f) different values of $\gamma_H$. Interval $t\in[0, 84]$ represents the period from June 2008 to December 2014

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Table 1. Descriptions and values of parameters in model (1)

Parameter	Interpretation	Value	Unit	Source
$\Lambda_{H}$	Recruiting of susceptible humans	$2.431\times10^{4}$	month$^{-1}$	[34]
$\mu_{H}$	Natural death rate of humans	$1.126\times10^{-3}$	month$^{-1}$	[34]
$a_{H}$	The baseline transmission rate	$8.00\times10^{-14}$	month$^{-1}$	Estimated
$b_{H}$	The magnitude of forcing	0.6	none	[54]
$\varphi_{H}$	The initial phase	$ 4.978$	none	Estimated
$\gamma_{H}$	Cure rate	0.131	month$^{-1}$	[44]
$\lambda_{M}$	Migration rate	209	month$^{-1}$	[5], [33]
$\mu_{M}$	Natural death rate of miracidia	27	month$^{-1}$	[18], [36]
$\Lambda_{V}$	Recruiting of susceptible snails	$5.660\times10^{5}$	month$^{-1}$	[8], [27], [53]
$\mu_{V}$	Natural death rate of snails	$ 1.788\times10^{-2}$	month$^{-1}$	[33]
$\alpha_{V}$	Disease induced death rate of snails	0.012	month$^{-1}$	[18], [33]
$a_{V}$	The baseline transmission rate	$1.974\times10^{-8}$	month$^{-1}$	Estimated
$b_{V}$	The magnitude of forcing	0.6	none	[54]
$\varphi_{V}$	The initial phase	$4.407$	none	Estimated
$\lambda_{P}$	Migration rate	78	month$^{-1}$	[18], [33]
$\mu_{P}$	Natural death rate of cercariae	0.12	month$^{-1}$	[18], [36]

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