Multi-host transmission dynamics of schistosomiasis and its optimal control

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2015, 12(5): 983-1006. doi: 10.3934/mbe.2015.12.983

Multi-host transmission dynamics of schistosomiasis and its optimal control

Chunxiao Ding ^1, , Zhipeng Qiu ^1, and Huaiping Zhu ^2,

1.	Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, 210094, China
2.	LAboratory of Mathematical Parallel Systems (LAMPS), Centre for Disease Modeling, Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3

Received December 2014 Revised March 2015 Published June 2015

Abstract
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In this paper we formulate a dynamical model to study the transmission dynamics of schistosomiasis in humans and snails. We also incorporate bovines in the model to study their impact on transmission and controlling the spread of Schistosoma japonicum in humans in China. The dynamics of the model is rigorously analyzed by using the theory of dynamical systems. The theoretical results show that the disease free equilibrium is globally asymptotically stable if $\mathcal R_0<1$, and if $\mathcal R_0>1$ the system has only one positive equilibrium. The local stability of the unique positive equilibrium is investigated and sufficient conditions are also provided for the global stability of the positive equilibrium. The optimal control theory are further applied to the model to study the corresponding optimal control problem. Both analytical and numerical results suggest that: (a) the infected bovines play an important role in the spread of schistosomiasis among humans, and killing the infected bovines will be useful to prevent transmission of schistosomiasis among humans; (b) optimal control strategy performs better than the constant controls in reducing the prevalence of the infected human and the cost for implementing optimal control is much less than that for constant controls; and (c) improving the treatment rate of infected humans, the killing rate of the infected bovines and the fishing rate of snails in the early stage of spread of schistosomiasis are very helpful to contain the prevalence of infected human case as well as minimize the total cost.

Keywords: optimal control., Multi-hosts, transmission dynamics, stability, schistosomiasis.

Mathematics Subject Classification: Primary: 92B05, 92D3.

Citation: Chunxiao Ding, Zhipeng Qiu, Huaiping Zhu. Multi-host transmission dynamics of schistosomiasis and its optimal control. Mathematical Biosciences & Engineering, 2015, 12 (5) : 983-1006. doi: 10.3934/mbe.2015.12.983

References:

[1]	A. Abdelrazec, S. Lenhart and H. Zhu, Transmission dynamics of West Nile virus in mosquitoes and corvids and non-corvids, Journal of Mathematical Biology, 68 (2014), 1553-1582. doi: 10.1007/s00285-013-0677-3.
[2]	L. J. Abu-Raddad, A. S. Magaret, C. Celum, A. Wald, I. M. Longini Jr, S. G. Self and L. Corey, Genital herpes has played a more important role than any other sexually transmitted infection in driving HIV prevalence in Africa, PloS One, 3 (2008), e2230. http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0002230
[3]	K. W. Blayneh, A. B. Gumel, S. Lenhart and C. Tim, Backward bifurcation and optimal control in transmission dynamics of West Nile virus, Bulletin of Mathematical Biology, 72 (2010), 1006-1028. doi: 10.1007/s11538-009-9480-0.
[4]	C. Castillo-Chevez and H. R. Thieme, Asymptotically autonomous epidemic models, Mathematical Population Dynamics: Analysis of Heterogeneity, 1 (1995), 33-50. http://www.researchgate.net/publication/221674057_Asymptotically_autonomous_epidemic_models
[5]	Z. Feng, C. Li and F. A. Milner, Schistosomiasis models with density dependence and age of infection in snail dynamics, Mathematical Biosciences, 177 (2002), 271-286. doi: 10.1016/S0025-5564(01)00115-8.
[6]	Z. Feng, Z. Qiu, Z. Sang, C. Lorenzo and J. Glasser, Modeling the synergy between HSV-2 and HIV and potential impact of HSV-2 therapy, Mathematical Biosciences, 245 (2013), 171-187. doi: 10.1016/j.mbs.2013.07.003.
[7]	A. Fenton and A. B. Pedersen, Community epidemiology framework for classifying disease threats, Emerging Infectious Diseases, 11 (2005), 1815-1821. http://wwwnc.cdc.gov/eid/article/11/12/05-0306_article
[8]	W. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, Springer, 1975. http://cds.cern.ch/record/1611958
[9]	D. J. Gray, G. M. Williams, Y. Li and D. P. McManus, Transmission dynamics of Schistosoma japonicum in the lakes and marshlands of China, PLoS One, 3 (2008), e4058. http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0004058
[10]	J. O. Lloyd-Smith, D. George, K. M. Pepin, V. E. Pitzer, J. R. Pulliam, A. P. Dobson, P. J. Hudson and B. T. Grenfell, Epidemic dynamics at the human-animal interface, Science, 326 (2009), 1362-1367, http://www.sciencemag.org/content/326/5958/1362.short
[11]	L. S. Pontryagin, Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley and Sons, Inc., New York-London, 1962.
[12]	M. Rafikov, L. Bevilacqua and A. P. P. Wyse, Optimal control strategy of malaria vector using genetically modified mosquitoes, Journal of Theoretical Biology, 258 (2009), 418-425. http://www.sciencedirect.com/science/article/pii/S0022519308004190 doi: 10.1016/j.jtbi.2008.08.006.
[13]	S. Riley, H. Carabin, P. Bélisle, L. Joseph, V. Tallo, E. Balolong, A. L. Willingham III, T. J. Fernandez Jr., R. O. Gonzales, R. Olveda and S. T. McGarvey, Multi-host transmission dynamics of Schistosoma japonicum in Samar Province, the Philippines, PLoS Medicine, 5 (2008), e18. http://dx.plos.org/10.1371/journal.pmed.0050018
[14]	J. W. Rudge, J. P. Webster, D. B. Lu, T. P. Wang, G. R. Fang and M. G. Basanez, Identifying host species driving transmission of schistosomiasis japonica, a multihost parasite system, in China, Proceedings of the National Academy of Sciences, 110 (2013), 11457-11462. http://www.pnas.org/content/110/28/11457.short
[15]	C. Shan, X. Zhou and H. Zhu, The Dynamics of Growing Islets and Transmission of Schistosomiasis Japonica in the Yangtze River, Bulletin of Mathematical Biology, 76 (2014), 1194-1217. doi: 10.1007/s11538-014-9961-7.
[16]	H. L. Smith, Cooperative systems of differential equations with concave nonlinearities, Nonlinear Analysis: Theory, Methods and Applications, 10 (1986), 1037-1052. http://www.sciencedirect.com/science/article/pii/0362546X86900878
[17]	H. L. Smith and P. Waltman, Perturbation of a globally stable steady state, Proceedings of the American Mathematical Society, 127 (1999), 447-453. doi: 10.1090/S0002-9939-99-04768-1.
[18]	P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[19]	W. Wang and X. Q. Zhao, An epidemic model in a patchy environment, Mathematical Biosciences, 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001.
[20]	World Health Organization, http://www.who.int/features/factfiles/schistosomiasis/en/.
[21]	M. J. Woolhouse, On the application of mathematical models of schistosome transmission dynamics. II. Control, Acta Tropica, 50 (1992), 189-204. http://www.sciencedirect.com/science/article/pii/0001706X9290076A
[22]	J. Xiang, H. Chen and H. Ishikawa, A mathematical model for the transmission of Schistosoma japonicum in consideration of seasonal water level fluctuations of Poyang Lake in Jiangxi, China, Parasitology International, 62 (2013), 118-126. http://www.sciencedirect.com/science/article/pii/S1383576912001341
[23]	P. Zhang, Z. Feng and F. Milner, A schistosomiasis model with an age-structure in human hosts and its application to treatment strategies, Mathematical Biosciences, 205 (2007), 83-107. doi: 10.1016/j.mbs.2006.06.006.
[24]	R. Zhao and F. A. Milner, A mathematical model of Schistosoma mansoni in Biomphalaria glabrata with control strategies, Bulletin of Mathematical Biology, 70 (2008), 1886-1905. doi: 10.1007/s11538-008-9330-5.
[25]	Y. B. Zhou, S. Liang and Q. W. Jiang, Factors impacting on progress towards elimination of transmission of schistosomiasis japonica in China, Parasit Vectors, 5 (2012), 257-275. http://www.biomedcentral.com/content/pdf/1756-3305-5-275.pdf

show all references

References:

[1]	A. Abdelrazec, S. Lenhart and H. Zhu, Transmission dynamics of West Nile virus in mosquitoes and corvids and non-corvids, Journal of Mathematical Biology, 68 (2014), 1553-1582. doi: 10.1007/s00285-013-0677-3.
[2]	L. J. Abu-Raddad, A. S. Magaret, C. Celum, A. Wald, I. M. Longini Jr, S. G. Self and L. Corey, Genital herpes has played a more important role than any other sexually transmitted infection in driving HIV prevalence in Africa, PloS One, 3 (2008), e2230. http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0002230
[3]	K. W. Blayneh, A. B. Gumel, S. Lenhart and C. Tim, Backward bifurcation and optimal control in transmission dynamics of West Nile virus, Bulletin of Mathematical Biology, 72 (2010), 1006-1028. doi: 10.1007/s11538-009-9480-0.
[4]	C. Castillo-Chevez and H. R. Thieme, Asymptotically autonomous epidemic models, Mathematical Population Dynamics: Analysis of Heterogeneity, 1 (1995), 33-50. http://www.researchgate.net/publication/221674057_Asymptotically_autonomous_epidemic_models
[5]	Z. Feng, C. Li and F. A. Milner, Schistosomiasis models with density dependence and age of infection in snail dynamics, Mathematical Biosciences, 177 (2002), 271-286. doi: 10.1016/S0025-5564(01)00115-8.
[6]	Z. Feng, Z. Qiu, Z. Sang, C. Lorenzo and J. Glasser, Modeling the synergy between HSV-2 and HIV and potential impact of HSV-2 therapy, Mathematical Biosciences, 245 (2013), 171-187. doi: 10.1016/j.mbs.2013.07.003.
[7]	A. Fenton and A. B. Pedersen, Community epidemiology framework for classifying disease threats, Emerging Infectious Diseases, 11 (2005), 1815-1821. http://wwwnc.cdc.gov/eid/article/11/12/05-0306_article
[8]	W. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, Springer, 1975. http://cds.cern.ch/record/1611958
[9]	D. J. Gray, G. M. Williams, Y. Li and D. P. McManus, Transmission dynamics of Schistosoma japonicum in the lakes and marshlands of China, PLoS One, 3 (2008), e4058. http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0004058
[10]	J. O. Lloyd-Smith, D. George, K. M. Pepin, V. E. Pitzer, J. R. Pulliam, A. P. Dobson, P. J. Hudson and B. T. Grenfell, Epidemic dynamics at the human-animal interface, Science, 326 (2009), 1362-1367, http://www.sciencemag.org/content/326/5958/1362.short
[11]	L. S. Pontryagin, Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley and Sons, Inc., New York-London, 1962.
[12]	M. Rafikov, L. Bevilacqua and A. P. P. Wyse, Optimal control strategy of malaria vector using genetically modified mosquitoes, Journal of Theoretical Biology, 258 (2009), 418-425. http://www.sciencedirect.com/science/article/pii/S0022519308004190 doi: 10.1016/j.jtbi.2008.08.006.
[13]	S. Riley, H. Carabin, P. Bélisle, L. Joseph, V. Tallo, E. Balolong, A. L. Willingham III, T. J. Fernandez Jr., R. O. Gonzales, R. Olveda and S. T. McGarvey, Multi-host transmission dynamics of Schistosoma japonicum in Samar Province, the Philippines, PLoS Medicine, 5 (2008), e18. http://dx.plos.org/10.1371/journal.pmed.0050018
[14]	J. W. Rudge, J. P. Webster, D. B. Lu, T. P. Wang, G. R. Fang and M. G. Basanez, Identifying host species driving transmission of schistosomiasis japonica, a multihost parasite system, in China, Proceedings of the National Academy of Sciences, 110 (2013), 11457-11462. http://www.pnas.org/content/110/28/11457.short
[15]	C. Shan, X. Zhou and H. Zhu, The Dynamics of Growing Islets and Transmission of Schistosomiasis Japonica in the Yangtze River, Bulletin of Mathematical Biology, 76 (2014), 1194-1217. doi: 10.1007/s11538-014-9961-7.
[16]	H. L. Smith, Cooperative systems of differential equations with concave nonlinearities, Nonlinear Analysis: Theory, Methods and Applications, 10 (1986), 1037-1052. http://www.sciencedirect.com/science/article/pii/0362546X86900878
[17]	H. L. Smith and P. Waltman, Perturbation of a globally stable steady state, Proceedings of the American Mathematical Society, 127 (1999), 447-453. doi: 10.1090/S0002-9939-99-04768-1.
[18]	P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[19]	W. Wang and X. Q. Zhao, An epidemic model in a patchy environment, Mathematical Biosciences, 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001.
[20]	World Health Organization, http://www.who.int/features/factfiles/schistosomiasis/en/.
[21]	M. J. Woolhouse, On the application of mathematical models of schistosome transmission dynamics. II. Control, Acta Tropica, 50 (1992), 189-204. http://www.sciencedirect.com/science/article/pii/0001706X9290076A
[22]	J. Xiang, H. Chen and H. Ishikawa, A mathematical model for the transmission of Schistosoma japonicum in consideration of seasonal water level fluctuations of Poyang Lake in Jiangxi, China, Parasitology International, 62 (2013), 118-126. http://www.sciencedirect.com/science/article/pii/S1383576912001341
[23]	P. Zhang, Z. Feng and F. Milner, A schistosomiasis model with an age-structure in human hosts and its application to treatment strategies, Mathematical Biosciences, 205 (2007), 83-107. doi: 10.1016/j.mbs.2006.06.006.
[24]	R. Zhao and F. A. Milner, A mathematical model of Schistosoma mansoni in Biomphalaria glabrata with control strategies, Bulletin of Mathematical Biology, 70 (2008), 1886-1905. doi: 10.1007/s11538-008-9330-5.
[25]	Y. B. Zhou, S. Liang and Q. W. Jiang, Factors impacting on progress towards elimination of transmission of schistosomiasis japonica in China, Parasit Vectors, 5 (2012), 257-275. http://www.biomedcentral.com/content/pdf/1756-3305-5-275.pdf

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