American Institute of Mathematical Sciences

Advanced
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.

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Centers for Disease Control and Prevention, Parasites – Schistosomiasis. Updated on November 7,2012. Available from: http://www.cdc.gov/parasites/schistosomiasis/biology.html.

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Chinese Center for Disease Control and Prevention, Schisosomiasis. Updated on November 11,2012. Available from: http://www.ipd.org.cn/Article/xxjs/hzdw/201206/2431.html.

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Chinese Center for Disease Control and Prevention/The Data-center of China Public Health Science, Schisosomiasis. Available from: http://www.phsciencedata.cn/Share/ky_sjml.jsp?id=5912cbb2-c84b-4bca-a554-7c234072a34c&show=0.

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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

N. Hairston, On the mathematical analysis of schistosome populations, Bull. WHO, 33 (1965), 45-62. 

[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.

[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. 

[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.

[29]

R. J. Larsen and M. L. Marx, An Introduction to Mathematical Statistics and Its Applications, 4$^{nd}$ edition, Pearson Education, 2012.

[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.

[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.

[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.

[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.

[34]

National Bureau of Statistics of China, China Demographic Yearbook of 2008. Available from: http://www.stats.gov.cn/tjsj/ndsj/2008/indexch.htm.

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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.

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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)
ParameterInterpretationValueUnitSource
$\Lambda_{H}$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 forcing0.6none[54]
$\varphi_{H}$The initial phase $ 4.978$noneEstimated
$\gamma_{H}$Cure rate0.131month$^{-1}$[44]
$\lambda_{M}$Migration rate209month$^{-1}$[5], [33]
$\mu_{M}$Natural death rate of miracidia27month$^{-1}$[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 snails0.012month$^{-1}$[18], [33]
$a_{V}$The baseline transmission rate $1.974\times10^{-8}$month$^{-1}$Estimated
$b_{V}$The magnitude of forcing0.6none[54]
$\varphi_{V}$The initial phase $4.407$noneEstimated
$\lambda_{P}$Migration rate78month$^{-1}$[18], [33]
$\mu_{P}$Natural death rate of cercariae0.12month$^{-1}$[18], [36]
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ParameterInterpretationValueUnitSource
$\Lambda_{H}$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 forcing0.6none[54]
$\varphi_{H}$The initial phase $ 4.978$noneEstimated
$\gamma_{H}$Cure rate0.131month$^{-1}$[44]
$\lambda_{M}$Migration rate209month$^{-1}$[5], [33]
$\mu_{M}$Natural death rate of miracidia27month$^{-1}$[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 snails0.012month$^{-1}$[18], [33]
$a_{V}$The baseline transmission rate $1.974\times10^{-8}$month$^{-1}$Estimated
$b_{V}$The magnitude of forcing0.6none[54]
$\varphi_{V}$The initial phase $4.407$noneEstimated
$\lambda_{P}$Migration rate78month$^{-1}$[18], [33]
$\mu_{P}$Natural death rate of cercariae0.12month$^{-1}$[18], [36]
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