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

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December 2017, 14(5&6): 1279-1299. doi: 10.3934/mbe.2017066

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

Yingke Li ^a,b, , Zhidong Teng ^a,, , Shigui Ruan ^c, , Mingtao Li ^d, and Xiaomei Feng ^e,

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)

Received April 05, 2016 Revised October 30, 2016 Published May 2017

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.

Keywords: Seasonal schistosomiasis model, basic reproduction number, extinction, uniform persistence.

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

Citation: Yingke Li, Zhidong Teng, Shigui Ruan, Mingtao Li, Xiaomei Feng. A mathematical model for the seasonal transmission of schistosomiasis in the lake and marshland regions of China. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1279-1299. doi: 10.3934/mbe.2017066

References:

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show all references

References:

[1]	S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual and P. Rohani, Seasonality and the dynamics of infectious diseases, Ecol. Lett., 9 (2006), 467-484. doi: 10.1111/j.1461-0248.2005.00879.x. Google Scholar
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[14]	M. Diaby, A. Iggidr, M. Sy and A. Sène, Global analysis of a schistosomiasis infection model with biological control, Appl. Math. Comput., 246 (2014), 731-742. doi: 10.1016/j.amc.2014.08.061. Google Scholar
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[16]	Z. Feng, A. Eppert, F. Milner and D. Minchella, Estimation of parameters governing the transmission dynamics of schistosomes, Appl. Math. Lett., 17 (2004), 1105-1112. doi: 10.1016/j.aml.2004.02.002. Google Scholar
[17]	Z. Feng, C. Li and F. Milner, Schistosomiasis models with density dependence and age of infection in snail dynamics, Math. Biosci., 177 (2002), 271-286. doi: 10.1016/S0025-5564(01)00115-8. Google Scholar
[18]	S. Gao, Y. Liu, Y. Luo and D. Xie, Control problems of mathematical model for schistosomiasis transmission dynamics, Nonlinear Dyn., 63 (2011), 503-512. doi: 10.1007/s11071-010-9818-z. Google Scholar
[19]	W. Garira, D. Mathebula and R. Netshikweta, A mathematical modelling framework for linked within-host and between-host dynamics for infections pathogens in the environment, Math. Biosci., 256 (2014), 58-78. doi: 10.1016/j.mbs.2014.08.004. Google Scholar
[20]	D. Gray, G. Williams, Y. Li and D. Mcmanus, Transmission dynamics of Schistosoma japonicum in the Lakes and Marshlands of China, PLoS One, 3 (2008), e4058. doi: 10.1371/journal.pone.0004058. Google Scholar
[21]	D. Gray, Y. Li, G. Williams, Z. Zhao, D. Harn, S. Li, M. Ren, Z. Feng, F. Guo, J. Guo, J. Zhou, Y. Dong, Y. Li, A. Ross and D. McManus, A multi-component integrated approach for the elimination of schistosomiasis in the People's Republic of China: Design and baseline results of a 4-year cluster-randomised intervention trial, Int. J. Parasitol., 44 (2014), 659-668. doi: 10.1016/j.ijpara.2014.05.005. Google Scholar
[22]	J. Greenman, M. Kamo and M. Boots, External forcing of ecological and epidemiological systems: A resonance approach, Physica D, 190 (2004), 136-151. doi: 10.1016/j.physd.2003.08.008. Google Scholar
[23]	B. Gryseels, K. Polman, J. Clerinx and L. Kestens, Human schistosomiasis, Lancet, 368 (2006), 1106-1118. doi: 10.1016/S0140-6736(06)69440-3. Google Scholar
[24]	A. Guiro, S. Ouaro and A. Traore, Stability analysis of a schistosomiasis model with delays, Adv. Differ. Equ., 2013 (2013), 15pp. doi: 10.1186/1687-1847-2013-303. Google Scholar
[25]	N. Hairston, On the mathematical analysis of schistosome populations, Bull. WHO, 33 (1965), 45-62. Google Scholar
[26]	G. Hu, J. Hu, K. Song, D. Lin, J. Zhang, C. Cao, J. Xu, D. Li and W. Jiang, The role of health education and health promotionin the control of schistosomiasis: experiences from a 12-year intervention study in the Poyang Lake area, Acta Trop., 96 (2005), 232-241. doi: 10.1016/j.actatropica.2005.07.016. Google Scholar
[27]	C. Huang, J. Zou, S. Li and X. Zhou, Survival and reproduction of Oncomelania hupensis robertsoni in water network regions in Hubei Province, China, Chin. J. Schisto. Control., 23 (2011), 173-177. Google Scholar
[28]	A. Hussein, I. Hassan and R. Khalifa, Development and hatching mechanism of Fasciola eggs, light and scanning electron microscopic studies, Saudi J. Biol. Sci., 17 (2010), 247-251. doi: 10.1016/j.sjbs.2010.04.010. Google Scholar
[29]	R. J. Larsen and M. L. Marx, An Introduction to Mathematical Statistics and Its Applications, 4$^{nd}$ edition, Pearson Education, 2012. Google Scholar
[30]	S. Liang, D. Maszle and R. Spear, A quantitative framework for a multi-group model of Schistosomiasis japonicum transmission dynamics and control in Sichuan China, Acta Trop., 82 (2002), 263-277. doi: 10.1016/S0001-706X(02)00018-9. Google Scholar
[31]	J. Liu, B. Peng and T. Zhang, Effect of discretization on dynamical behavior of SEIR and SIR models with nonlinear incidence, Appl Math Lett, 39 (2015), 60-66. doi: 10.1016/j.aml.2014.08.012. Google Scholar
[32]	G. Macdonald, The dynamics of helminth infections, with special reference to schistosomes, Trans. R. Soci. Trop. Med. Hyg., 59 (1965), 489-506. doi: 10.1016/0035-9203(65)90152-5. Google Scholar
[33]	T. Mangal, S. Paterson and A. Fenton, Predicting the impact of long-term temperature changes on the epidemiology and control of schistosomiasis: a mechanistic model. PLoSOne., 3 (2008), e1438. doi: 10.1371/journal.pone.0001438. Google Scholar
[34]	National Bureau of Statistics of China, China Demographic Yearbook of 2008. Available from: http://www.stats.gov.cn/tjsj/ndsj/2008/indexch.htm. Google Scholar
[35]	M. Rios, J. Garcia, J. Sanchez and D. Perez, A statistical analysis of the seasonality in pulmonary tuberculosis, Eur. J. Epidemiol., 16 (2000), 483-488. doi: 10.1023/A:1007653329972. Google Scholar
[36]	R. Spear, A. Hubbard, S. Liang and E. Seto, Disease transmission models for public health decision making: Toward an approach for designing intervention strategies for Schistosomiasis japonica, Environ. Health Perspect., 110 (2002), 907-915. Google Scholar
[37]	L. Sun, X. Zhou, Q. Hong, G. Yang, Y. Huang, W. Xi and Y. Jiang, Impact of global warming on transmission of schistosomiasis in China Ⅲ. Relationship between snail infections rate and environmental temperature, Chin.J.Schist. Control, 15 (2003), 161-163. Google Scholar
[38]	Z. Teng and L. Chen, The positive periodic solutions of periodic Kolmogorove type systems with delays, Acta Math. Appl. Sin., 22 (1999), 446-456. Google Scholar
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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 5">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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Table 1">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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Table 1">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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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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American Institute of Mathematical Sciences

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

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

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