# American Institute of Mathematical Sciences

December  2014, 19(10): 3133-3145. doi: 10.3934/dcdsb.2014.19.3133

## A periodic Ross-Macdonald model in a patchy environment

 1 Francis I. Proctor Foundation for Research in Ophthalmology, University of California, San Francisco, San Francisco, CA 94143, United States 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong 3 Department of Mathematics, University of Miami, Coral Gables, FL 33124

Received  July 2013 Revised  January 2014 Published  October 2014

Based on the classical Ross-Macdonald model, in this paper we propose a periodic malaria model to incorporate the effects of temporal and spatial heterogeneity on disease transmission. The temporal heterogeneity is described by assuming that some model coefficients are time-periodic, while the spatial heterogeneity is modeled by using a multi-patch structure and assuming that individuals travel among patches. We calculate the basic reproduction number $\mathcal{R}_0$ and show that either the disease-free periodic solution is globally asymptotically stable if $\mathcal{R}_0\le 1$ or the positive periodic solution is globally asymptotically stable if $\mathcal{R}_0>1$. Numerical simulations are conducted to confirm the analytical results and explore the effect of travel control on the disease prevalence.
Citation: Daozhou Gao, Yijun Lou, Shigui Ruan. A periodic Ross-Macdonald model in a patchy environment. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3133-3145. doi: 10.3934/dcdsb.2014.19.3133
##### 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. [2] G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis, Math. Biosci., 38 (1973), 113-122. doi: 10.1016/0025-5564(78)90021-4. [3] P. Auger, E. Kouokam, G. Sallet, M. Tchuente and B. Tsanou, The Ross-Macdonald model in a patchy environment, Math. Biosci., 216 (2008), 123-131. doi: 10.1016/j.mbs.2008.08.010. [4] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0. [5] N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population, Bull. Math. Biol., 69 (2007), 1067-1091. doi: 10.1007/s11538-006-9166-9. [6] Z. Bai and Y. Zhou, Threshold dynamics of a Bacillary Dysentery model with seasonal fluctuation, Discrete Contin. Dyn. Syst., Ser. B, 15 (2011), 1-14. doi: 10.3934/dcdsb.2011.15.1. [7] C. Cosner, J. C. Beier, R. S. Cantrell, D. Impoinvil, L. Kapitanski, M. D. Potts, A. Troyo and S. Ruan, The effects of human movement on the persistence of vector-borne diseases, J. Theoret. Biol., 258 (2009), 550-560. doi: 10.1016/j.jtbi.2009.02.016. [8] C. Costantini, S. G. Li, A. D. Torre, N. Sagnon, M. Coluzzi and C. E. Taylor, Density, survival and dispersal of Anopheles gambiae complex mosquitoes in a West African Sudan savanna village, Med. Vet. Entomol., 10 (1996), 203-219. doi: 10.1111/j.1365-2915.1996.tb00733.x. [9] M. H. Craig, I. Kleinschmidt, J. B. Nawn, D. Le Sueur and B. L. Sharp, Exploring 30 years of malaria case data in KwaZulu-Natal, South Africa: Part I. The impact of climatic factors, Trop. Med. Int. Health, 9 (2004), 1247-1257. doi: 10.1111/j.1365-3156.2004.01340.x. [10] B. Dembele, A. Friedman and A.-A. Yakubu, Malaria model with periodic mosquito birth and death rates, J. Biol. Dynam., 3 (2009), 430-445. doi: 10.1080/17513750802495816. [11] D. Gao and S. Ruan, A multi-patch malaria model with logistic growth populations, SIAM J. Appl. Math., 72 (2012), 819-841. doi: 10.1137/110850761. [12] H. Gao, L. Wang, S. Liang, Y. Liu, S. Tong, J. Wang, Y. Li, X. Wang, H. Yang, J. Ma, L. Fang and W. Cao, Change in rainfall drives malaria re-emergence in Anhui province, China, PLoS ONE, 7 (2012), e43686. doi: 10.1371/journal.pone.0043686. [13] N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology, Proc. R. Soc. B., 273 (2006), 2541-2550. doi: 10.1098/rspb.2006.3604. [14] J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1980. [15] M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems, J. Dyn. Differ. Equ., 13 (2001), 107-131. doi: 10.1023/A:1009044515567. [16] K. D. Lafferty, The ecology of climate change and infectious diseases, Ecology, 90 (2009), 888-900. doi: 10.1890/08-0079.1. [17] L. Liu, X.-Q. Zhao and Y. Zhou, A tuberculosis model with seasonality, Bull. Math. Biol., 72 (2010), 931-952. doi: 10.1007/s11538-009-9477-8. [18] X. Liu and X.-Q. Zhao, A periodic epidemic model with age structure in a patchy environment, SIAM J. Appl. Math., 71 (2011), 1896-1917. doi: 10.1137/100813610. [19] J. Lou, Y. Lou and J. Wu, Threshold virus dynamics with impulsive antiretroviral drug effects, J. Math. Biol., 65 (2012), 623-652. doi: 10.1007/s00285-011-0474-9. [20] Y. Lou and X.-Q. Zhao, The periodic Ross-Macdonald model with diffusion and advection, Appl. Anal., 89 (2010), 1067-1089. doi: 10.1080/00036810903437804. [21] Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), 2023-2044. doi: 10.1137/080744438. [22] Y. Lou and X.-Q. Zhao, Modelling malaria control by introduction of larvivorous fish, Bull. Math. Biol., 73 (2011), 2384-2407. doi: 10.1007/s11538-011-9628-6. [23] G. Macdonald, The analysis of sporozoite rate, Trop. Dis. Bull., 49 (1952), 569-585. [24] G. Macdonald, Epidemiological basis of malaria control, Bull. World Health Organ., 15 (1956), 613-626. [25] G. Macdonald, The Epidemiology and Control of Malaria, Oxford University Press, London, 1957. [26] J. T. Midega, C. M. Mbogo, H. Mwambi, M. D. Wilson, G. Ojwang, J. M. Mwangangi, J. G. Nzovu, J. I. Githure, G. Yan and J. C. Beier, Estimating dispersal and survival of Anopheles gambiae and Anopheles funestus along the Kenyan coast by using mark-release-recapture methods, J. Med. Entomol., 44 (2007), 923-929. doi: {10.1603/0022-2585(2007)44[923:EDASOA]2.0.CO;2}. [27] R. S. Ostfeld, Climate change and the distribution and intensity of infectious diseases, Ecology, 90 (2009), 903-905. doi: 10.1890/08-0659.1. [28] P. E. Parham and E. Michael, Modeling the effects of weather and climate change on malaria transmission, Environ. Health Perspect., 118 (2010), 620-626. doi: 10.1289/ehp.0901256. [29] D. Rain, Eaters of the Dry Season: Circular Labor Migration in the West African Sahel, Westview Press, Boulder, CO, 1999. [30] A. Roca-Feltrer, J. R. Armstrong Schellenberg, L. Smith and I. Carneiro, A simple method for defining malaria seasonality, Malaria J., 8 (2009), 276. doi: 10.1186/1475-2875-8-276. [31] R. Ross, The Prevention of Malaria, 2nd edn., Murray, London, 1911. [32] S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114. doi: 10.1007/s11538-007-9292-z. [33] D. L. Smith, J. Dushoff and F. E. McKenzie, The risk of a mosquito-borne infection in a heterogeneous environment, PLoS Biology, 2 (2004), 1957-1964. [34] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, vol. 41, A.M.S., Providence, RI, 1995. [35] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043. [36] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equ., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. [37] P. Weng and X.-Q. Zhao, Spatial dynamics of a nonlocal and delayed population model in a periodic habitat, Discrete Contin. Dyn. Syst. Ser. A, 29 (2011), 343-366. doi: 10.3934/dcds.2011.29.343. [38] J. Zhang, Z. Jin, G. Sun, X. Sun and S. Ruan, Modeling seasonal Rabies epidemics in China, Bull. Math. Biol., 74 (2012), 1226-1251. doi: 10.1007/s11538-012-9720-6. [39] T. Zhang and Z. Teng, On a nonautonomous SEIRS model in epidemiology, Bull. Math. Biol., 69 (2007), 2537-2559. doi: 10.1007/s11538-007-9231-z. [40] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.

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##### 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. [2] G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis, Math. Biosci., 38 (1973), 113-122. doi: 10.1016/0025-5564(78)90021-4. [3] P. Auger, E. Kouokam, G. Sallet, M. Tchuente and B. Tsanou, The Ross-Macdonald model in a patchy environment, Math. Biosci., 216 (2008), 123-131. doi: 10.1016/j.mbs.2008.08.010. [4] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0. [5] N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population, Bull. Math. Biol., 69 (2007), 1067-1091. doi: 10.1007/s11538-006-9166-9. [6] Z. Bai and Y. Zhou, Threshold dynamics of a Bacillary Dysentery model with seasonal fluctuation, Discrete Contin. Dyn. Syst., Ser. B, 15 (2011), 1-14. doi: 10.3934/dcdsb.2011.15.1. [7] C. Cosner, J. C. Beier, R. S. Cantrell, D. Impoinvil, L. Kapitanski, M. D. Potts, A. Troyo and S. Ruan, The effects of human movement on the persistence of vector-borne diseases, J. Theoret. Biol., 258 (2009), 550-560. doi: 10.1016/j.jtbi.2009.02.016. [8] C. Costantini, S. G. Li, A. D. Torre, N. Sagnon, M. Coluzzi and C. E. Taylor, Density, survival and dispersal of Anopheles gambiae complex mosquitoes in a West African Sudan savanna village, Med. Vet. Entomol., 10 (1996), 203-219. doi: 10.1111/j.1365-2915.1996.tb00733.x. [9] M. H. Craig, I. Kleinschmidt, J. B. Nawn, D. Le Sueur and B. L. Sharp, Exploring 30 years of malaria case data in KwaZulu-Natal, South Africa: Part I. The impact of climatic factors, Trop. Med. Int. Health, 9 (2004), 1247-1257. doi: 10.1111/j.1365-3156.2004.01340.x. [10] B. Dembele, A. Friedman and A.-A. Yakubu, Malaria model with periodic mosquito birth and death rates, J. Biol. Dynam., 3 (2009), 430-445. doi: 10.1080/17513750802495816. [11] D. Gao and S. Ruan, A multi-patch malaria model with logistic growth populations, SIAM J. Appl. Math., 72 (2012), 819-841. doi: 10.1137/110850761. [12] H. Gao, L. Wang, S. Liang, Y. Liu, S. Tong, J. Wang, Y. Li, X. Wang, H. Yang, J. Ma, L. Fang and W. Cao, Change in rainfall drives malaria re-emergence in Anhui province, China, PLoS ONE, 7 (2012), e43686. doi: 10.1371/journal.pone.0043686. [13] N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology, Proc. R. Soc. B., 273 (2006), 2541-2550. doi: 10.1098/rspb.2006.3604. [14] J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1980. [15] M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems, J. Dyn. Differ. Equ., 13 (2001), 107-131. doi: 10.1023/A:1009044515567. [16] K. D. Lafferty, The ecology of climate change and infectious diseases, Ecology, 90 (2009), 888-900. doi: 10.1890/08-0079.1. [17] L. Liu, X.-Q. Zhao and Y. Zhou, A tuberculosis model with seasonality, Bull. Math. Biol., 72 (2010), 931-952. doi: 10.1007/s11538-009-9477-8. [18] X. Liu and X.-Q. Zhao, A periodic epidemic model with age structure in a patchy environment, SIAM J. Appl. Math., 71 (2011), 1896-1917. doi: 10.1137/100813610. [19] J. Lou, Y. Lou and J. Wu, Threshold virus dynamics with impulsive antiretroviral drug effects, J. Math. Biol., 65 (2012), 623-652. doi: 10.1007/s00285-011-0474-9. [20] Y. Lou and X.-Q. Zhao, The periodic Ross-Macdonald model with diffusion and advection, Appl. Anal., 89 (2010), 1067-1089. doi: 10.1080/00036810903437804. [21] Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), 2023-2044. doi: 10.1137/080744438. [22] Y. Lou and X.-Q. Zhao, Modelling malaria control by introduction of larvivorous fish, Bull. Math. Biol., 73 (2011), 2384-2407. doi: 10.1007/s11538-011-9628-6. [23] G. Macdonald, The analysis of sporozoite rate, Trop. Dis. Bull., 49 (1952), 569-585. [24] G. Macdonald, Epidemiological basis of malaria control, Bull. World Health Organ., 15 (1956), 613-626. [25] G. Macdonald, The Epidemiology and Control of Malaria, Oxford University Press, London, 1957. [26] J. T. Midega, C. M. Mbogo, H. Mwambi, M. D. Wilson, G. Ojwang, J. M. Mwangangi, J. G. Nzovu, J. I. Githure, G. Yan and J. C. Beier, Estimating dispersal and survival of Anopheles gambiae and Anopheles funestus along the Kenyan coast by using mark-release-recapture methods, J. Med. Entomol., 44 (2007), 923-929. doi: {10.1603/0022-2585(2007)44[923:EDASOA]2.0.CO;2}. [27] R. S. Ostfeld, Climate change and the distribution and intensity of infectious diseases, Ecology, 90 (2009), 903-905. doi: 10.1890/08-0659.1. [28] P. E. Parham and E. Michael, Modeling the effects of weather and climate change on malaria transmission, Environ. Health Perspect., 118 (2010), 620-626. doi: 10.1289/ehp.0901256. [29] D. Rain, Eaters of the Dry Season: Circular Labor Migration in the West African Sahel, Westview Press, Boulder, CO, 1999. [30] A. Roca-Feltrer, J. R. Armstrong Schellenberg, L. Smith and I. Carneiro, A simple method for defining malaria seasonality, Malaria J., 8 (2009), 276. doi: 10.1186/1475-2875-8-276. [31] R. Ross, The Prevention of Malaria, 2nd edn., Murray, London, 1911. [32] S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114. doi: 10.1007/s11538-007-9292-z. [33] D. L. Smith, J. Dushoff and F. E. McKenzie, The risk of a mosquito-borne infection in a heterogeneous environment, PLoS Biology, 2 (2004), 1957-1964. [34] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, vol. 41, A.M.S., Providence, RI, 1995. [35] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043. [36] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equ., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. [37] P. Weng and X.-Q. Zhao, Spatial dynamics of a nonlocal and delayed population model in a periodic habitat, Discrete Contin. Dyn. Syst. Ser. A, 29 (2011), 343-366. doi: 10.3934/dcds.2011.29.343. [38] J. Zhang, Z. Jin, G. Sun, X. Sun and S. Ruan, Modeling seasonal Rabies epidemics in China, Bull. Math. Biol., 74 (2012), 1226-1251. doi: 10.1007/s11538-012-9720-6. [39] T. Zhang and Z. Teng, On a nonautonomous SEIRS model in epidemiology, Bull. Math. Biol., 69 (2007), 2537-2559. doi: 10.1007/s11538-007-9231-z. [40] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.
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