Modeling the effect of temperature variability on malaria control strategies

Salisu M. Garba¹ ; Usman A. Danbaba¹

¹ Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa.

Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 65

Résumé

In this study, a non-autonomous (temperature dependent) and autonomous (temperature independent) models for the transmission dynamics of malaria in a population are designed and rigorously analysed. The models are used to assess the impact of temperature changes on various control strategies. The autonomous model is shown to exhibit the phenomenon of backward bifurcation, where an asymptotically-stable disease-free equilibrium (DFE) co-exists with an asymptotically-stable endemic equilibrium when the associated reproduction number is less than unity. This phenomenon is shown to arise due to the presence of imperfect vaccines and disease-induced mortality rate. Threshold quantities (such as the basic offspring number, vaccination and host type reproduction numbers) and their interpretations for the models are presented. Conditions for local asymptotic stability of the disease-free solutions are computed. Sensitivity analysis using temperature data obtained from Kwazulu Natal Province of South Africa [K. Okuneye and A.B. Gumel. Mathematical Biosciences 287 (2017) 72–92] is used to assess the parameters that have the most influence on malaria transmission. The effect of various control strategies (bed nets, adulticides and vaccination) were assessed via numerical simulations.

Détail

DOI : 10.1051/mmnp/2020044

Bibliographie

[1] A. Abdelrazec, A.B. Gumel Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics J. Math. Biol 2017 1351 1395

[2] F.B. Agusto, A.B. Gumel, P.E. Parham Qualitative assessment of the role of temperature variations on malaria transmission dynamics J. Biolog. Syst 2015 1550030

[3] D. Alonso, M.J. Bouma, M. Pascual Epidemic malaria and warmer temperatures in recent decades in an East African highland Proc. Roy. Soc. Lond. B 2011 1661 1669

[4] R. Anguelov, Y. Dumont, J. Lubuma Mathematical modeling of sterile insect technology for control of anopheles mosquito Comput. Math. Appl 2012 374 389

[5] J.I. Blanford, S. Blanford, R.G. Crane, M.E. Mann, K.P. Paaijmans, K.V. Schreiber, M.B. Thomas Implications of temperature variation for malaria parasite development across Africa Sci. Rep 2013 1300

[6] J.F. Briere, P. Pracros, A.Y. Le Roux, J.S. Pierre A novel rate model of temperature-dependent development for arthropods Environ. Entomol 1999 22 29

[7] T.M. Bury, C.T. Bauch, M. Anand Charting pathways to climate change mitigation in a coupled socio-climate model PLoS Comput. Biol 2019 e1007000

[8] C. Caminade, S. Kovats, J. Rocklov, A.M. Tompkins, A.P. Morse, F.J. Colón-González, H. Stenlund, P. Martens, S.J. Lloyd Impact of climate change on global malaria distribution Proc. Natl. Acad. Sci 2014 3286 3291

[9] C. Castillo-Chavez, B. Song Dynamical models of tuberculosis and their applications Math. Biosci. Eng 2004 361 404

[10] N. Chitnis, J.M. Cushing, J.M. Hyman Bifurcation analysis of a mathematical model for malaria transmission SIAM J. Appl. Math 2006 24 45

[11] U.A. Danbaba and S.M. Garba, Analysis of model for the transmission dynamics of Zika with sterile insect technique. In: Mathematical Methods and Models in Biosciences, edited by R. Anguelov, M. Lachowicz. Biomath Forum, Sofia, GNU General Public License, University British Columbia, Canada (2018) 81–99. https://doi.org/10.11145/texts.2018.01.083.

[12] U.A. Danbaba, S.M. Garba Modeling the transmission dynamics of Zika with sterile insect technique Math. Methods Appl. Sci 2018 8871 8896

[13] B. Dembele, A. Friedman, A.A. Yakubu Malaria model with periodic mosquito birth and death rates J. Biol. Dyn 2009 430 445

[14] Y. Dumont, F. Chiroleu, C. Domerg On a temporal model for the Chikungunya disease: modeling, theory and numerics Math. Biosci 2008 80 91

[15] Y. Dumont, F. Chiroleu Vector control for the Chikungunya disease Math. Biosci. Eng 2010 313 345

[16] J. Dushoff, J.B. Plotkin, S.A. Levin, D.J. Earn Dynamical resonance can account for seasonality of influenza epidemics Proc. Natl. Acad. Sci 2004 16915 16916

[17] S.E. Eikenberry, A.B. Gumel Mathematical modeling of climate change and malaria transmission dynamics: a historical review J. Math. Biol 2018 1 77

[18] S.M. Garba, A.B. Gumel, M.A. Bakar Backward bifurcations in dengue transmission dynamics Math. Biosci 2008 11 25

[19] S.M. Garba, A.B. Gumel Effect of cross-immunity on the transmission dynamics of two strains of dengue Int. J. Comput. Math 2010 2361 2384

[20] S.M. Garba, M.A. Safi Mathematical analysis of West Nile virus model with discrete delays Acta Math. Sci 2013 1439 1462

[21] D. Greenhalgh, I.A. Moneim SIRS epidemic model and simulations using different types of seasonal contact rate Syst. Anal. Model. Simul 2010 573 600

[22] P.W. Gething, T.P. Van Boeckel, D.L. Smith, C.A. Guerra, A.P. Patil, R.W. Snow, S.I. Hay Modelling the global constraints of temperature on transmission of Plasmodium falciparum and P. vivax Parasites Vectors 2011 92

[23] J.A. Heesterbeek, M.G. Roberts The type-reproduction number T in models for infectious disease control Math. Biosci 2007 3 10

[24] V. Laperriere, K. Brugger, F. Rubel Simulation of the seasonal cycles of bird, equine and human West Nile virus cases Prevent. Veter. Med 2011 99 110

[25] Y. Lou, X.Q. Zhao A climate-based malaria transmission model with structured vector population SIAM J. Appl. Math 2010 2023 2044

[26] I.A. Moneim, Greenhalgh Use of a periodic vaccination strategy to control the spread of epidemics with seasonally varying contact rate Math. Biosci. Eng 2005 591 611

[27] E.A. Mordecai, K.P. Paaijmans, L.R. Johnson, C. Balzer, T. Ben-Horin, E. De Moor, A. Mcnally, S. Pawar, S.J. Ryan, T.C. Smith, K.D. Lafferty Optimal temperature for malaria transmission is dramatically lower than previously predicted Ecol. Lett 2013 22 30

[28] C.C. Murdock, E.D. Sternberg, M.B. Thomas Malaria transmission potential could be reduced with current and future climate change Sci. Rep 2016 27771

[29] G.A. Ngwa, W.S. Shu A mathematical model for endemic malaria with variable human and mosquito populations Math. Comput. Model 2000 747 763

[30] A.M. Niger, A.B. Gumel Mathematical analysis of the role of repeated exposure on malaria transmission dynamics Differ. Equ. Dyn. Syst 2008 251 287

[31] K. Okuneye, A.B. Gumel Analysis of a temperature-and rainfall-dependent model for malaria transmission dynamics Math. Biosci 2017 72 92

[32] K. Okuneye, S.E. Eikenberry, A.B. Gumel Weather-driven malaria transmission model with gonotrophic and sporogonic cycles J. Biol. Dyn 2019 288 324

[33] S. Olaniyi, K.O. Okosun, O. Adesanya and E.A. Areo, Global stability and optimal control analysis of malaria dynamics in the presence of human travelers. 10 (2018) 166–186.

[34] P.E. Parham, J. Waldock, G.K. Christophides, D. Hemming, F. Agusto, K.J. Evans, N. Fefferman, H. Gaff, A. Gumel, S. Ladeau, S. Lenhart Climate, environmental and socio-economic change: weighing up the balance in vector-borne disease transmission Philo. Trans. R. Soc. B 2015 20130551

[35] M.A. Penny, R. Verity, C.A. Bever, C. Sauboin, K. Galactionova, S. Flasche, M.T. White, E.A. Wenger, N. Van De Velde, P. Pemberton-Ross, J.T. Griffin Public health impact and cost-effectiveness of the RT, S/AS01 malaria vaccine: a systematic comparison of predictions from four mathematical models The Lancet 2016 367 375

[36] F. Rubel, K. Brugger, M. Hantel, S. Chvala-Mannsberger, T. Bakonyi, H. Weissenböck, N. Nowotny Explaining Usutu virus dynamics in Austria: model development and calibration Prev. Veter. Med 2008 166 186

[37] J. Shaman, M. Spiegelman, M. Cane, M. Stieglitz A hydrologically driven model of swamp water mosquito population dynamics Ecol. Model 2006 395 404

[38] R. Tuteja Malaria-an overview FEBS J 2007 4670 4679

[39] P. Van Den Driessche, J. Watmough Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission Math. Biosci. 2002 29 48

[40] W. Wang, X.Q. Zhao Threshold dynamics for compartmental epidemic models in periodic environments J. Dyn. Differ. Equ 2008 699 717

[41] M.T. White, R. Verity, T.S. Churcher, A.C. Ghani Vaccine approaches to malaria control and elimination: Insights from mathematical models Vaccine 2015 7544 7550

[42] Malaria: World Health Organization fact-sheets. Available at http://www.who.int/news-room/fact-sheets/detail/malaria accessed on 1st August (2018).

[43] H.M. Yang A mathematical model for malaria transmission relating global warming and local socioeconomic conditions Rev. Saude Pub 2001 224 231