On the delayed vector-bias malaria model in an almost periodic environment

Bing He; Lizhong Qiang

doi:10.3934/dcdsb.2023010

Article Contents

2023, Volume 28, Issue 8: 4261-4280. Doi: 10.3934/dcdsb.2023010

This issue Previous Article Next Article Strong convergence rate for slow-fast stochastic differential equations with Markovian switching

On the delayed vector-bias malaria model in an almost periodic environment

Bing He^1, , and
Lizhong Qiang^2,

1.
Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China
2.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

^*Corresponding author: Bing He
^*Corresponding author: Bing He

Received: August 2022

Revised: November 2022

Early access: February 2023

Published online: February 2023

The authors are supported by the China Scholarship Council, NSF of China (Nos. 12071491, 12101501), Natural Science Foundation of Gansu Province (No. 21JR7RA150) and Guangzhou Science and Technology Plan Project (No. 202102080177).

Abstract / Introduction Full Text(HTML) Figure(4) Related Papers Cited by

Abstract

The global dynamics of a vector-bias malaria model with extrinsic incubation period in an almost periodic environment are considered. The basic reproduction ratio $ R_{0} $ is first introduced, and then it is proved that $ R_{0} $ is a threshold parameter that determines the global dynamics of the model, that is, the disease-free almost periodic solution is globally attractive when $ R_{0} $ $ <1 $, the system allows a unique positive almost periodic solution which has global attractivity when $ R_{0} $ $ >1 $. Numerical simulations show that extending the incubation period is advantageous for disease control, and the basic reproduction ratio may be overestimated or underestimated if vector bias is ignored.

Keywords:

Mathematics Subject Classification: Primary: 34D45, 37B55; Secondary: 92D30.

Citation:

Full Text(HTML)

Figure 1. The solution trajectory of (1) for $R_{0}>1$..

Download: Full-size image PowerPoint slide

Figure 2. The solution trajectory of (1) for $R_{0}<1$..

Download: Full-size image PowerPoint slide

Figure 3. The relationship between $R_{0}$ and $r$..

Download: Full-size image PowerPoint slide

Figure 4. The vector-bias effect..

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	Z. Bai, R. Peng and X.-Q. Zhao, A reaction-diffusion malaria model with seasonality and incubation period, J. Math. Biol., 77 (2018), 201-228. doi: 10.1007/s00285-017-1193-7.
[2]	P. H. Bezandry and T. Diagana, Almost Periodic Stochastic Processes, Springer, New York, 2011. doi: 10.1007/978-1-4419-9476-9.
[3]	F. Chamchod and N. F. Britton, Analysis of a vector-bias model on malaria transmission, Bull. Math. Biol., 73 (2011), 639-657. doi: 10.1007/s11538-010-9545-0.
[4]	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.
[5]	C. Corduneanu, Almost Periodic Oscillations and Waves, Springer, New York, 2009. doi: 10.1007/978-0-387-09819-7.
[6]	T. Diagana, S. Elaydi and A.-Z. Yakubu, Population models in almost periodic environments, J. Difference Equ. Appl., 13 (2007), 239-260. doi: 10.1080/10236190601079035.
[7]	A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Vol. 377. Springer-Verlag, Berlin-New York, 1974.
[8]	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.
[9]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988. doi: 10.1090/surv/025.
[10]	J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.
[11]	M. A. Ibrahim and A. D$\acute{e}$nes, Threshold and stability results in a periodic model for malaria transmission with partial immunity in humans, Appl. Math. Comput., 392 (2021), 125711. doi: 10.1016/j.amc.2020.125711.
[12]	J. G. Kingsolver, Mosquito host choice and the epidemiology of malaria, Am. Nat., 130 (1987), 811-827. doi: 10.1086/284749.
[13]	R. Lacroix, W. R. Mukabana, L. C. Gouagna and J. C. Koella, Malaria infection increases attractiveness of humans to mosquitoes, PLOS Biol., 3 (2005), 1590-1593. doi: 10.1371/journal.pbio.0030298.
[14]	X. Liang, L. Zhang and X.-Q. Zhao, Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease), J. Dyn. Differ. Equ., 31 (2019), 1247-1278. doi: 10.1007/s10884-017-9601-7.
[15]	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.
[16]	Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568. doi: 10.1007/s00285-010-0346-8.
[17]	G. Macdonald, The Epidemiology and Control of Malaria, Oxford University Press, London, 1957.
[18]	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.
[19]	S. Novo, R. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dyn. Differ. Equ., 25 (2013), 1201-1231. doi: 10.1007/s10884-013-9337-y.
[20]	S. Novo, R. Obaya and A. M. Sanz, Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows, Nonlinearity, 26 (2013), 2409-2440. doi: 10.1088/0951-7715/26/9/2409.
[21]	K. P. Paaijmans, L. J. Cator and M. B. Thomas, Temperature-dependent pre-bloodmeal period and temperature-driven asynchrony between parasite development and mosquito biting rate reduce malaria transmission intensity, PLoS ONE., 8 (2013), e55777. doi: 10.1371/journal.pone.0055777.
[22]	L. Qiang and B.-G. Wang, An almost periodic malaria transmission model with time-delayed input of vector, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1525-1546. doi: 10.3934/dcdsb.2017073.
[23]	L. Qiang, B.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental models with time delay, J. Differential Equations, 269 (2020), 4440-4476. doi: 10.1016/j.jde.2020.03.027.
[24]	L. Qiang, B.-G. Wang and X.-Q. Zhao, A stage-structured population model with time-dependent delay in an almost periodic environment, J. Dyn. Differ. Equ., 34 (2022), 341-364. doi: 10.1007/s10884-020-09827-6.
[25]	R. Ross, The Prevention of Malaria, 2nd edn. Murray, London, 1911.
[26]	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.
[27]	G. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold, London, 1971
[28]	W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Memoirs of Amer. Math. Soc., 136 (1998), x+93 pp. doi: 10.1090/memo/0647.
[29]	H. L. Smith, Monotone Dynamics Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, RI, 1995.
[30]	H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870.
[31]	J. Wang and Y. Chen, Threshold dynamics of a vector-borne disease model with spatial structure and vector-bias, Appl. Math. Lett., 100 (2020), 106052. doi: 10.1016/j.aml.2019.106052.
[32]	X. Wang and X.-Q. Zhao, A periodic vector-bias malaria model with incubation period, SIAM J. Appl. Math., 77 (2017), 181-201. doi: 10.1137/15M1046277.
[33]	X. Wang and X.-Q. Zhao, A malaria transmission model with temperature-dependent incubation period, Bull. Math. Biol., 79 (2017), 1155-1182. doi: 10.1007/s11538-017-0276-3.
[34]	R. Wu and X.-Q. Zhao, A reaction-diffusion model of vector-borne disease with periodic delays, J. Nonlinear Sci., 29 (2019), 29-64. doi: 10.1007/s00332-018-9475-9.
[35]	Z. Xu and X.-Q. Zhao, A vector-bias malaria model with incubation period and diffusion, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2615-2634. doi: 10.3934/dcdsb.2012.17.2615.
[36]	T. Yang and L. Zhang, Remarks on basic reproduction ratios for periodic abstract functional differential equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6771-6782. doi: 10.3934/dcdsb.2019166.
[37]	C. Zhang, Almost Periodic Type Functions and Ergodicity, Science Press, Beijing; Kluwer Academic Publishers, Dordrecht, 2003.
[38]	R. Zhang and J. Wang, On the global attractivity for a reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 84 (2022), Paper No. 53, 12 pp. doi: 10.1007/s00285-022-01751-1.
[39]	H. Zhao, Y. Shi and X. Zhang, Dynamic analysis of a malaria reaction-diffusion model with periodic delays and vector bias, Math. Biosci. Eng., 19 (2022), 2538-2574. doi: 10.3934/mbe.2022117.
[40]	X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Differ. Equ., 29 (2017), 67-82. doi: 10.1007/s10884-015-9425-2.
[41]	X.-Q. Zhao, Dynamical Systems in Population Biology, 2nd edn. Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.