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Morse indices and symmetry breaking for the Gelfand equation in expanding annuli
An almost periodic malaria transmission model with time-delayed input of vector
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China |
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.
References:
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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. |
[2] |
J. L. Aron,
Mathematical modeling of immunity to malaria, Math. Biosci., 90 (1988), 385-396.
doi: 10.1016/0025-5564(88)90076-4. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
C. Corduneanu,
Almost Periodic Functions Chelsea Publishing Company New York, N. Y. , 1989. |
[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. |
[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. |
[11] |
A. M. Fink,
Almost Periodic Differential Equations Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1974. |
[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. |
[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. |
[14] |
J. K. Hale,
Asymptotic Behavior of Dissipative Systems Math. Surveys and Monographs 25, Amer. Math. Soc. , Providence, RI, 1988. |
[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. |
[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. |
[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. |
[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. |
[19] |
G. Macdonald et al,
The Epidemiology and Control of Malaria Oxford University Press, Oxford, UK, 1957. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
P. Reiter,
Climate change and mosquito-borne disease, Envir. Hlth. Perspect., 109 (2001), 141-161.
doi: 10.2307/3434853. |
[26] |
R. Ross,
The Prevention of Malaria John Murray, London, 1911. |
[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. |
[28] |
G. Sell,
Topological Dynamics and Ordinary Differential Equations Van Nostrand Reinhold, London, 1971. |
[29] |
M. Service,
Mosquito Ecology: Field Sampling Methods Springer Netherlands, 1993. |
[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. |
[31] |
H. L. Smith and P. Waltman,
The Theory of the Chemostat Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511530043. |
[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. |
[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. |
[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. |
[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. |
[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. |
[37] |
The world health report, 2014. Available from: http://www.who.int/malaria/media/en/. |
[38] |
X. -Q. Zhao,
Dynamical Systems in Population Biology Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
[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. |
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. |
[2] |
J. L. Aron,
Mathematical modeling of immunity to malaria, Math. Biosci., 90 (1988), 385-396.
doi: 10.1016/0025-5564(88)90076-4. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
C. Corduneanu,
Almost Periodic Functions Chelsea Publishing Company New York, N. Y. , 1989. |
[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. |
[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. |
[11] |
A. M. Fink,
Almost Periodic Differential Equations Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1974. |
[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. |
[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. |
[14] |
J. K. Hale,
Asymptotic Behavior of Dissipative Systems Math. Surveys and Monographs 25, Amer. Math. Soc. , Providence, RI, 1988. |
[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. |
[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. |
[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. |
[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. |
[19] |
G. Macdonald et al,
The Epidemiology and Control of Malaria Oxford University Press, Oxford, UK, 1957. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
P. Reiter,
Climate change and mosquito-borne disease, Envir. Hlth. Perspect., 109 (2001), 141-161.
doi: 10.2307/3434853. |
[26] |
R. Ross,
The Prevention of Malaria John Murray, London, 1911. |
[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. |
[28] |
G. Sell,
Topological Dynamics and Ordinary Differential Equations Van Nostrand Reinhold, London, 1971. |
[29] |
M. Service,
Mosquito Ecology: Field Sampling Methods Springer Netherlands, 1993. |
[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. |
[31] |
H. L. Smith and P. Waltman,
The Theory of the Chemostat Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511530043. |
[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. |
[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. |
[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. |
[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. |
[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. |
[37] |
The world health report, 2014. Available from: http://www.who.int/malaria/media/en/. |
[38] |
X. -Q. Zhao,
Dynamical Systems in Population Biology Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
[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. |




Parameters | Meaning |
Infection probability from infective humans to susceptible mosquitoes. | |
Transfer rate of mosquitoes from the exposed class to the infectious class. | |
| The proportion between the probability of transmission from recovered humans to susceptible vectors and the probability of transmission from infectious humans. |
| Per capita loss rate of immunity for humans. |
Transfer ratio of humans from the exposed class to the infectious class. | |
| Human recovery rate. |
Infection probability from infectious mosquitoes to susceptible humans. | |
| Death rate of humans. |
The diseased-induced rate of humans. | |
Supplement rate of humans. |
Parameters | Meaning |
Infection probability from infective humans to susceptible mosquitoes. | |
Transfer rate of mosquitoes from the exposed class to the infectious class. | |
| The proportion between the probability of transmission from recovered humans to susceptible vectors and the probability of transmission from infectious humans. |
| Per capita loss rate of immunity for humans. |
Transfer ratio of humans from the exposed class to the infectious class. | |
| Human recovery rate. |
Infection probability from infectious mosquitoes to susceptible humans. | |
| Death rate of humans. |
The diseased-induced rate of humans. | |
Supplement rate of humans. |
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