American Institute of Mathematical Sciences

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

A periodic Ross-Macdonald model in a patchy environment

Daozhou Gao 1, , Yijun Lou 2, and  Shigui Ruan 3,

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.
Keywords: basic reproduction number., patch model, threshold dynamics, Malaria, seasonality.
Mathematics Subject Classification: Primary: 92D30; Secondary: 34D2.
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.

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.

[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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