Abstract
Insecticide-treated bed nets (ITNs) are among the most important and effective intervention measures against malaria. In order to investigate the impact of bed net use on disease control, we formulate a periodic vector-bias malaria model incorporating the juvenile stage of mosquitoes and the use of ITNs. We derive the vector reproduction ratio \(R_v\) and the basic reproduction ratio \(R_0\). We show that the global dynamics of the model is completely determined by these two reproduction ratios. More precisely, the mosquito-free periodic solution is globally attractive if \(R_v<1\); the unique disease-free periodic solution is globally attractive if \(R_v>1\) and \(R_0<1\); and the model admits a unique positive periodic solution and it is globally attractive if \(R_v>1\) and \(R_0>1\). Numerically, we study the malaria transmission case in Port Harcourt, Nigeria. Our findings show that the use of ITNs has a positive effect on reducing \(R_0\), and that malaria may be eliminated from this area if over 75% of the human population were to use ITNs. The simulation about the long term behavior of solutions has good agreement with the obtained analytic result. Moreover, we find that the ignorance of the vector-bias effect may result in underestimation of the basic reproduction ratio \(R_0\). Another notable result is that the infection risk would be underestimated if the basic reproduction ratio \([R_0]\) of the time-averaged autonomous system were used.
References
Abebe A, Abebel G, Tsegaye W, Golassa L (2011) Climatic variables and malaria transmission dynamics in Jimma town, South West Ethiopia. Parasit Vectors 4(30):1–11
Ai S, Li J, Lu J (2012) Mosquito-stage-structured malaria models and their global dynamics. SIAM J Appl Math 72(4):1213–1237
Agusto FB, Del Valle SY, Blayneh KW, Ngonghala CN, Goncalves MJ, Li N, Zhao R, Gong H (2013) The impact of bed-net use on malaria prevalence. J Theor Biol 320:58–65
Arino J, Ducrot A, Zongo P (2012) A metapopulation model for malaria with transmission-blocking partial immunity in hosts. J Math Biol 64:423–448
Aron JL, May RM (1982) The population dynamics of malaria. In: Anderson RM (ed) The population dynamics of infectious diseases: theory and applications. Chapman and Hall, London, pp 139–179
Bacaër N, Ait Dads EH (2012) On the biological interpretation of a definition for the parameter \(R_0\) in periodic population models. J Math Biol 65:601–621
Bacaër N, Guernaoui S (2006) The epidemic threshold of vector-borne diseases with seasonality. J Math Biol 53:421–436
Birget PLG, Koella JC (2015) An epidemiological model of the effects of insecticide-treated bed nets on malaria transmission. PLoS ONE 10(12):e0144173. doi:10.1371/journal.pone.0144173
Bowman C, Gumel AB, van den Driessche P, Wu J, Zhu H (2005) A mathematical model for assessing control strategies against West Nile virus. Bull Math Biol 67:1107–1133
Chamchod F, Britton NF (2011) Analysis of a vector-bias model on malaria transmission. Bull Math Biol 73:639–657
Chitnis N, Hyman JM, Cushing JM (2008) Determining important parameters in the spread of malaria through the sensitivity anaysis of a mathematical model. Bull Math Biol 70:1272–1296
Chitnis N, Schapira A, Smith T, Steketee R (2010) Comparing the effectiveness of malaria vector-control interventions through a mathematical model. Am J Trop Med Hyg 83(2):230–240
D’Alessandro U, LOlaleye BO, McGuire W, Langercock P, Bennet S (1995) Mortality and morbidity from malaria in Gambian children after introduction of an impregnated bednet programme. Lancet 345:479–483
Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio \(R_0\) in the models for infectious disease in heterogeneous populations. J Math Biol 28:365–382
George IO, Jeremiah I, Kasso T (2013) Prevalence of congenital malaria in Port Harcourt, Nigeria. Br J Med Med Res 3(2):398–406
Hale JK, Verduyn Lunel SM (1993) Introduction to functional differential equations. Springer, New York
Hirsch MW, Smith HL, Zhao X-Q (2001) Chain transitivity, attractivity, and strong repellors for semifynamical systems. J Dyn Differ Equ 13:107–131
Inaba H (2012) On a new perspective of the basic reproduction number in heterogeneous environments. J Math Biol 22:113–128
Kesavan SK, Reddy NP (1985) On the feeding strategy and the mechanics of blood sucking in insects. J Theor Biol 113:781–783
Killeen GF, Smith TA (2007) Exploring the contributions of bed nets, cattle, insecticides and excitorepellency to malaria control: a deterministic model of mosquito host-seeking behaviour and mortality. Trans R Soc Trop Med Hyg 101(9):867–880
Kingsolver JG (1987) Mosquito host choice and the epidemiology of malaria. Am Nat 130:811–827
Koella JC (1991) On the use of mathematical models of malaria transmission. Acta Trop 49:1–25
Lacroix R, Mukabana WR, Gouagna LC, Koella JC (2005) Malaria infection increases attractiveness of humans to mosquitoes. PLoS Biol 3:e298
Lengeler C (2004) Insecticide-treated nets for malaria control: real gains, Bull WHO, pp 82–84
Li J, Welch RM, Nair US, Sever TL, Irwin DE, Cordon-Rosales C, Padilla N (2002) Dynamic malaria models with environmental changes. In: Proceedings of the thirty-fourth southeastern symposium on system theory, Huntsville, AL, pp 396–400
Li J (2009) Simple stage-structured models for wild and transgenic mosquito populations. J Differ Equ Appl 15:327–47
Lou Y, Zhao X-Q (2010) A climate-based malaria transmission model with structured vector population. SIAM J Appl Math 70(6):2023–2044
Lou Y, Zhao X-Q (2011) Modelling malaria control by introduction of larvivorous fish. Bull Math Biol 73:2384–2407
Macdonald G (1957) The epidemiology and control of malaria. Oxford University Press, London
Martens P, Niessen LW, Rotmans J, Jetten TH, McMichael AJ (1995) Potential impact of global climate change on malaria risk. Environ Health Perspect 103(5):458–464
Ngarakana-Gwasira ET, Bhunu CP, Mashonjowa E (2014) Assessing the impact of temperature on malaria transmission dynamics. Afr Mat 25:1095–1112
Ngonghala CN, Del Valle SY, Zhao R, Mohammed-Awel J (2014) Quantifying the impact of decay in bed-net efficacy on malaria transmission. J Theor Biol 363:247–261
Ngonghala CN, Mohammed-Awel J, Zhao R, Prosper O (2016) Interplay between insecticide-treated bed-nets and mosquito demography: implications for malaria control. J Theor Biol 397:179–192
Paaijmans KP, Cator LJ, Thomas MB (2009) Temperature-dependent pre-bloodmeal period and temperature-driven asynchrony between parasite development and mosquito biting rate reduce malaria transmission intensity. PLoS ONE 8(1):e55777
Reiskind MH, Lounibos LP (2009) Effects of intraspecific larval competition on adult longevity in the mosquitoes Aedes aegypti and Aedes albopictus. Med Vet Entomol 23:62–68
Ross R (1911) The prevention of malaria, 2nd edn. Murray, London
Rubel F, Brugger K, Hantel M, Chvala-Mannsberger S, Bakonyi T, Weissenbo H, Nowotny N (2008) Explaining Usutu virus dynamics in Austria: model development and calibration. Prev Vet Med 85:166186
Smith HL (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, mathematical surveys and monographs, vol 41. American Mathematical Society, Providence
Thieme HR (2009) Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J Appl Math 70:188–211
Uneke CJ (2009) Impact of home management of Plasmodium falciparum malaria on childhood malaria control in sub-Saharan Africa. Trop Biomed 26:182–199
van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48
Walter W (1997) On strongly monotone flows. In: Annales Polonici Mathematici vol LXVI, pp 269–274
Wang C, Gourley SA, Liu R (2014) Delayed action insecticides and their role in mosquito and malaria control. J Math Biol 68:417–451
Wang W, Zhao X-Q (2008) Threshold dynamics for compartmental epidemic models in periodic environments. J Dyn Differ Equ 20:699–717
Wang X, Zhao X-Q (2017) A periodic vector-bias malaria model with incubation period. SIAM J Appl Math 77(1):181–201
World Health Organisation (WHO) (2015) Global malaria programme, World Malaria report
Wonham MJ, de Camino-Beck T, Lewis MA (2004) An epidemiological model for West Nile Virus: Invasion analysis and control applications. Proc R Soc Lond B Biol Sci 271:501–507
Xu Z, Zhao X-Q (2012) A vector-bias malaria model with incubation period and diffusion. Discrete Continuous Dyn Syst Ser B 17(7):2615–2634
Yakob L, Yan G (2009) Modeling the effects of integrating larval habitat source reduction and insecticide treated nets for malaria control. PLoS ONE 4(9):e6921. doi:10.1371/journal.pone.0006921
Zhao X-Q (2003) Dynamical systems in population biology. Springer, New York
Zhao X-Q (2017) Basic reproduction ratios for periodic compartmental models with time delay. J Dyn Differ Equ 29:67–82
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This work is supported in part by the NSERC of Canada.
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Wang, X., Zhao, XQ. A climate-based malaria model with the use of bed nets. J. Math. Biol. 77, 1–25 (2018). https://doi.org/10.1007/s00285-017-1183-9
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DOI: https://doi.org/10.1007/s00285-017-1183-9