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:
[1]

S. AltizerA. DobsonP. HosseiniP. HudsonM. 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

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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

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M. CraigI. KleinschmidtJ. NawnD. 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

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O. DiekmannJ. 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

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A. M. Fink, Almost Periodic Differential Equations Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1974.  Google Scholar

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D. GaoY. 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

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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. JepsonA. 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-GwasiraC. 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. RuanD. 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. WangW. 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. AltizerA. DobsonP. HosseiniP. HudsonM. 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. ChitnisJ. 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. ChiyakaW. 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. ChiyakaJ. M. TchuencheW. 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. CraigI. KleinschmidtJ. NawnD. 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. DiekmannJ. 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. GaoY. 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. JepsonA. 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-GwasiraC. 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. RuanD. 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. WangW. 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

Figure 1.  Compartmental model for malaria.
Figure 2.  The total number of vector.
Figure 3.  The graph of $\|u(t)\|$ and $\ln\|u(t)\|$ of (26) when $k=2.5$.
Figure 4.  The long-term behavior of vector and human populations when $R_{0}<1$.
Figure 5.  The graph of $\|u(t)\|$ and $\ln\|u(t)\|$ of (26) when $k=5$.
Figure 6.  The long-term behavior of vector and human populations when $R_{0}>1$.
Figure 7.  Relationship between $\widehat{p}$ and $R_{0}$.
Figure 8.  The long-term trend of $\ln\|u(t)\|$ of (27) with $m=2.4$ and $m=2.5$.
Table 1.  The parameters of model and their means.
ParametersMeaning
$a$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.
ParametersMeaning
$a$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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