# American Institute of Mathematical Sciences

June  2017, 22(4): 1525-1546. doi: 10.3934/dcdsb.2017073

## An almost periodic malaria transmission model with time-delayed input of vector

 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

* Corresponding author:Bin-Guo Wang

Received  April 2016 Revised  December 2016 Published  February 2017

An almost periodic malaria transmission model with the time-delayed input of vector is considered. It is shown that the disease is uniformly persistent when the basic reproduction ratio $R_{0}>1$, and it will die out when $R_{0} < 1$ under the assumption that there exists a small invasion. Furthermore, the global stability of the disease-free almost periodic state is obtained provided that the disease-induced death rate is null. Finally, we illustrate the above results by numerical simulations and show that the periodic epidemic models may overestimate or underestimate the malaria risk.

Citation: Lizhong Qiang, Bin-Guo Wang. An almost periodic malaria transmission model with time-delayed input of vector. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1525-1546. doi: 10.3934/dcdsb.2017073
##### References:
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Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173. Google Scholar [21] J. Nedelman, Introductory review some new thoughts about some old malaria models, Math. Biosci., 73 (1985), 159-182. doi: 10.1016/0025-5564(85)90010-0. Google Scholar [22] E. Ngarakana-Gwasira, C. Bhunu and E. Mashonjowa, Assessing the impact of temperature on malaria transmission dynamics, Afr. Mat., 25 (2014), 1095-1112. doi: 10.1007/s13370-013-0178-y. Google Scholar [23] G. Ngwa, Modelling the dynamics of endemic malaria in growing populations, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1173-1202. doi: 10.3934/dcdsb.2004.4.1173. Google Scholar [24] G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Math. Comput. Modelling, 32 (2000), 747-763. doi: 10.1016/S0895-7177(00)00169-2. Google Scholar [25] P. Reiter, Climate change and mosquito-borne disease, Envir. Hlth. Perspect., 109 (2001), 141-161. doi: 10.2307/3434853. Google Scholar [26] R. Ross, The Prevention of Malaria John Murray, London, 1911.Google Scholar [27] 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. Google Scholar [28] G. Sell, Topological Dynamics and Ordinary Differential Equations Van Nostrand Reinhold, London, 1971. Google Scholar [29] M. Service, Mosquito Ecology: Field Sampling Methods Springer Netherlands, 1993.Google Scholar [30] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative systems Math. Surveys and Monographs, 41, Amer. Math. Soc. , Providence, RI, 1995. Google Scholar [31] H. L. Smith and P. Waltman, The Theory of the Chemostat Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043. Google Scholar [32] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [33] H. Wan and H. Zhu, The impact of resource and temperature on malaria transmission, Journal of Biological Systems, 20 (2012), 285-302. doi: 10.1142/S0218339012500118. Google Scholar [34] B. G. Wang, W. T. Li and L. Zhang, An almost periodic epidemic model with age structure in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 291-311. doi: 10.3934/dcdsb.2016.21.291. Google Scholar [35] B.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models, J. Dyn. Diff. Equ., 25 (2013), 535-562. doi: 10.1007/s10884-013-9304-7. Google Scholar [36] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. Google Scholar [37] The world health report, 2014. Available from: http://www.who.int/malaria/media/en/.Google Scholar [38] X. -Q. Zhao, Dynamical Systems in Population Biology Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1. Google Scholar [39] X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems, J. Differential Equations, 187 (2003), 494-509. doi: 10.1016/S0022-0396(02)00054-2. Google Scholar

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, Ecology Letters, 9 (2006), 467-484. doi: 10.1111/j.1461-0248.2005.00879.x. Google Scholar [2] J. L. Aron, Mathematical modeling of immunity to malaria, Math. Biosci., 90 (1988), 385-396. doi: 10.1016/0025-5564(88)90076-4. Google Scholar [3] A. Bomblies, Modeling the role of rainfall patterns in seasonal malaria transmission, Climatic Change, 112 (2012), 673-685. doi: 10.1007/s10584-011-0230-6. Google Scholar [4] N. Chitnis and J. M. Hyman, Bifurcation analysis of a mathematical model for malaria transmission, SIAM J. Appl. Math., 67 (2006), 24-45. doi: 10.1137/050638941. Google Scholar [5] N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296. doi: 10.1007/s11538-008-9299-0. Google Scholar [6] C. Chiyaka, W. Garira and S. Dube, Transmission model of endemic human malaria in a partially immune population, Math. Comput. Modelling, 46 (2007), 806-822. doi: 10.1016/j.mcm.2006.12.010. Google Scholar [7] C. Chiyaka, J. M. Tchuenche, W. Garira and S. Dube, A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria, Appl. Math. Comput., 195 (2008), 641-662. doi: 10.1016/j.amc.2007.05.016. Google Scholar [8] C. Corduneanu, Almost Periodic Functions Chelsea Publishing Company New York, N. Y. , 1989.Google Scholar [9] M. Craig, I. Kleinschmidt, J. Nawn, D. Le Sueur and B. 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. Google Scholar [10] O. Diekmann, J. A. P. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio $R_{0}$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. Google Scholar [11] A. M. Fink, Almost Periodic Differential Equations Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1974. Google Scholar [12] D. Gao, Y. Lou and S. Ruan, A periodic ross-macdonald model in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3133-3145. doi: 10.3934/dcdsb.2014.19.3133. Google Scholar [13] H. -W. Gao, L. -P. Wang, S. Liang, Y. -X. Liu, S. -L. Tong, J. -J. Wang, Y. -P. Li, X. -F. Wang, H. Yang and J. -Q. Ma, et al. , Change in rainfall drives malaria re-emergence in anhui province china, PLoS ONE, 7 (2012), e43686. doi: 10.1371/journal.pone.0043686. Google Scholar [14] J. K. Hale, Asymptotic Behavior of Dissipative Systems Math. Surveys and Monographs 25, Amer. Math. Soc. , Providence, RI, 1988. Google Scholar [15] M. B. Hoshen and A. P. Morse, A weather-driven model of malaria transmission, Malar. J., 3 (2004), 32-46. doi: 10.1186/1475-2875-3-32. Google Scholar [16] W. Jepson, A. Moutia and C. Courtois, The malaria problem in mauritius: The bionomics of mauritian anophelines, Bulletin of entomological research, 38 (1947), 177-208. doi: 10.1017/S0007485300030273. Google Scholar [17] Y. Lou and X.-Q. Zhao, Threshold dynamics in a time-delayed periodic SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 169-186. doi: 10.3934/dcdsb.2009.12.169. Google Scholar [18] 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. Google Scholar [19] G. Macdonald et al, The Epidemiology and Control of Malaria Oxford University Press, Oxford, UK, 1957.Google Scholar [20] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173. Google Scholar [21] J. Nedelman, Introductory review some new thoughts about some old malaria models, Math. Biosci., 73 (1985), 159-182. doi: 10.1016/0025-5564(85)90010-0. Google Scholar [22] E. Ngarakana-Gwasira, C. Bhunu and E. Mashonjowa, Assessing the impact of temperature on malaria transmission dynamics, Afr. Mat., 25 (2014), 1095-1112. doi: 10.1007/s13370-013-0178-y. Google Scholar [23] G. Ngwa, Modelling the dynamics of endemic malaria in growing populations, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1173-1202. doi: 10.3934/dcdsb.2004.4.1173. Google Scholar [24] G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Math. Comput. Modelling, 32 (2000), 747-763. doi: 10.1016/S0895-7177(00)00169-2. Google Scholar [25] P. Reiter, Climate change and mosquito-borne disease, Envir. Hlth. Perspect., 109 (2001), 141-161. doi: 10.2307/3434853. Google Scholar [26] R. Ross, The Prevention of Malaria John Murray, London, 1911.Google Scholar [27] 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. Google Scholar [28] G. Sell, Topological Dynamics and Ordinary Differential Equations Van Nostrand Reinhold, London, 1971. Google Scholar [29] M. Service, Mosquito Ecology: Field Sampling Methods Springer Netherlands, 1993.Google Scholar [30] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative systems Math. Surveys and Monographs, 41, Amer. Math. Soc. , Providence, RI, 1995. Google Scholar [31] H. L. Smith and P. Waltman, The Theory of the Chemostat Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043. Google Scholar [32] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [33] H. Wan and H. Zhu, The impact of resource and temperature on malaria transmission, Journal of Biological Systems, 20 (2012), 285-302. doi: 10.1142/S0218339012500118. Google Scholar [34] B. G. Wang, W. T. Li and L. Zhang, An almost periodic epidemic model with age structure in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 291-311. doi: 10.3934/dcdsb.2016.21.291. Google Scholar [35] B.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models, J. Dyn. Diff. Equ., 25 (2013), 535-562. doi: 10.1007/s10884-013-9304-7. Google Scholar [36] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. Google Scholar [37] The world health report, 2014. Available from: http://www.who.int/malaria/media/en/.Google Scholar [38] X. -Q. Zhao, Dynamical Systems in Population Biology Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1. Google Scholar [39] X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems, J. Differential Equations, 187 (2003), 494-509. doi: 10.1016/S0022-0396(02)00054-2. Google Scholar
Compartmental model for malaria.
The total number of vector.
The graph of $\|u(t)\|$ and $\ln\|u(t)\|$ of (26) when $k=2.5$.
The long-term behavior of vector and human populations when $R_{0}<1$.
The graph of $\|u(t)\|$ and $\ln\|u(t)\|$ of (26) when $k=5$.
The long-term behavior of vector and human populations when $R_{0}>1$.
Relationship between $\widehat{p}$ and $R_{0}$.
The long-term trend of $\ln\|u(t)\|$ of (27) with $m=2.4$ and $m=2.5$.
The parameters of model and their means.
 Parameters Meaning $a$ Infection probability from infective humans to susceptible mosquitoes. $\eta$ Transfer rate of mosquitoes from the exposed class to the infectious class. $\lambda$ The proportion between the probability of transmission from recovered humans to susceptible vectors and the probability of transmission from infectious humans. $\gamma_{R}$ Per capita loss rate of immunity for humans. $\gamma_{E}$ Transfer ratio of humans from the exposed class to the infectious class. $\gamma_{I}$ Human recovery rate. $c$ Infection probability from infectious mosquitoes to susceptible humans. $d_{h}$ Death rate of humans. $\sigma_{h}$ The diseased-induced rate of humans. $\Lambda_{h}$ Supplement rate of humans.
 Parameters Meaning $a$ Infection probability from infective humans to susceptible mosquitoes. $\eta$ Transfer rate of mosquitoes from the exposed class to the infectious class. $\lambda$ The proportion between the probability of transmission from recovered humans to susceptible vectors and the probability of transmission from infectious humans. $\gamma_{R}$ Per capita loss rate of immunity for humans. $\gamma_{E}$ Transfer ratio of humans from the exposed class to the infectious class. $\gamma_{I}$ Human recovery rate. $c$ Infection probability from infectious mosquitoes to susceptible humans. $d_{h}$ Death rate of humans. $\sigma_{h}$ The diseased-induced rate of humans. $\Lambda_{h}$ Supplement rate of humans.
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