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Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China. (English) Zbl 1356.92081
Summary: In this paper, we consider a deterministic malaria transmission model with standard incidence rate and treatment. Human population is divided into susceptible, infectious and recovered subclasses, and mosquito population is split into susceptible and infectious classes. It is assumed that, among individuals with malaria who are treated or recovered spontaneously, a proportion moves to the recovered class with temporary immunity and the other proportion returns to the susceptible class. Firstly, it is shown that two endemic equilibria may exist when the basic reproduction number $$\mathcal{R}_0 < 1$$ and a unique endemic equilibrium exists if $$\mathcal{R}_0 > 1 .$$ The presence of a backward bifurcation implies that it is possible for malaria to persist even if $$\mathcal{R}_0 < 1$$. Secondly, using geometric method, some sufficient conditions for global stability of the unique endemic equilibrium are obtained when $$\mathcal{R}_0 > 1$$. To deal with this problem, the estimate of the Lozinskiĭ measure of a $$6 \times 6$$ matrix is discussed. Finally, numerical simulations are provided to support our theoretical results. The model is also used to simulate the human malaria data reported by the Chinese Ministry of Health from 2002 to 2013. It is estimated that the basic reproduction number $$\mathcal{R}_0 \approx 0.0161$$ for the malaria transmission in China and it is found that the plan of eliminating malaria in China is practical under the current control strategies.

##### MSC:
 92D30 Epidemiology 34C23 Bifurcation theory for ordinary differential equations 34D20 Stability of solutions to ordinary differential equations
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##### References:
 [1] Snow, R. W.; Guerra, C. A.; Noor, A. M.; Myint, H. Y.; Hay, S. J., The global distribution of clinical episodes of plasmodium falciparum malaria, Nature, 434, 214, (2005) [2] Gallup, J. L.; Sachs, J. D., The economic burden of malaria, Am. J. Trop. Med. Hyg., 64, 85, (2001) [3] World Health Organization (WHO), 10 facts on malaria, http://www.who.int/features/factfiles/malaria/en (accessed 15.05.13). [4] World Health Organization (WHO), New report signals slowdown in the fight against malaria, http://www.who.int/mediacentre/news/releases/2012/malaria_20121217/en/index.html (accessed 17.12.12). [5] World Health Organization (WHO), Malaria, Fact sheet No.94, http://www.who.int/mediacentre/factsheets/fs094/en (accessed 15.03.13). [6] National Health and Family Planning Commission of the People’s Republic of China (NHFPC), http://www.nhfpc.gov.cn/jkj/index.shtml. [7] Xiangyang Daily, “Shaking chills” were history, March 17, 2009, http://www.xfrb.hj.cn/Read.asp?NewsID=132314. [8] Baike, Malaria, http://www.baike.com/wikdoc/sp/qr/history/version.do?ver=27&hisiden=q,eXEAAAgEA2F6,eXB6dQADDQ. [9] National Health and Family Planning Commission of the People’s Republic of China (NHFPC), Action Plans for the Elimination of Malaria (2010-2020) question and answer, http://www.nhfpc.gov.cn/zhuzhan/jbyfykz/201304/8b4bf70ebbff4148b4e9721806d20a64.shtml. [10] Li, B., China’s three serious challenges in the prevention and treatment of infectious diseases, China Pharm., 25, 3413, (2013) [11] Diebner, H. H.; Eincher, M.; Molineaux, L.; Collins, W. E.; Jeffert, G. M.; Dietz, K., Modelling the transition of asexual blood stages of plasmodium falciparum to gametocytes, J. Theoret. Biol., 202, 113-127, (2000) [12] White, N. J.; Pukrittayakamee, S.; Hien, T. T.; Faiz, M. A.; Mokuolu, O. A.; Dondorp, A. M., Malaria, Lancet, 383, 723-735, (2014) [13] R. Ross, 1911, The Prevention of Malaria, London, Murry. [14] Macdonald, G., The analysis of infection rates in diseases in which superinfection occurs, Trop. Dis. Bull., 47, 907-915, (1950) [15] Macdonald, G., The Epidemiology and Control of Malaria, (1957), Oxford University Press Oxford [16] N. Bailey, 1982, The Biomathematics of Malaria, London, Charles Griff. · Zbl 0494.92018 [17] Aron, J. L.; May, R. M., The population dynamics of malaria, (Anderson, R. M., (1982), Chapman & Hall London), 139-179 [18] Koella, J. C., On the use of mathematical models of malaria transmission, Acta Trop., 49, 1-25, (1991) [19] Ngwa, G. A.; Shu, W. S., A mathematical model for endemic malaria with variable human and mosquito populations, Math. Comput. Model., 32, 747-763, (2000) · Zbl 0998.92035 [20] Ngwa, G. A., Modelling the dynamics of endemic malaria in growing population, Discr. Cont. Dyn. Syst. B, 4, 1173-1202, (2004) · Zbl 1052.92048 [21] Chitnis, N.; Cushing, J. M.; Hyman, J. M., Bifurcation analysis of a mathematical model for malaria transmission, SIAM J. Appl. Math., 67, 24-45, (2006) · Zbl 1107.92047 [22] Wan, H.; Cui, J.-A., A model for the transmission of malaria, Discr. Cont. Dyn. Syst. B, 11, 479-496, (2009) · Zbl 1153.92028 [23] Tumwiine, J.; Mugisha, J.; Luboobi, L., A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity, Appl. Math. Comput., 189, 1953-1965, (2007) · Zbl 1117.92039 [24] Tumwiine, J.; Mugisha, J.; Luboobi, L., A host-vector model for malaria with infective immigrants, J. Math. Anal. Appl., 361, 139-149, (2010) · Zbl 1176.92045 [25] Chamchod, F.; Britton, N. F., Analysis of a vector-bias model on malaria transmission, Bull. Math. Biol., 73, 639-657, (2011) · Zbl 1225.92030 [26] Vargas-De-León, C., Global analysis of a delayed vector-bias model for malaria transmission with incubation period in mosquitoes, Math. Biosci. Eng., 9, 165-174, (2012) · Zbl 1259.34071 [27] Wang, L.; Teng, Z.; Zhang, T., Threshold dynamics of a malaria transmission model in periodic environment, Commun. Nonl. Sci. Numer. Simulat., 18, 1288-1303, (2013) · Zbl 1274.92050 [28] Agusto, F. B.; DelValle, S. Y.; Blayneh, K. W.; Ngonghala, C. N.; Goncalves, M. J.; Li, N., The impact of bed-net use on malaria prevalence, J. Theor. Biol., 320, 58-65, (2013) · Zbl 1406.92549 [29] Okosun, K. O.; Ouifki, R.; Marcus, N., Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity, Biosyst., 106, 136-145, (2011) [30] Buonomo, B.; Vargas-De-León, C., Stability and bifurcation analysis of a vector-bias model of malaria transmission, Math. Biosci., 242, 59-67, (2013) · Zbl 1316.92081 [31] Ngonghala, C. N.; Ngwa, G. A.; Teboh-Ewungkemc, M. I., Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission, Math. Biosci., 240, 45-62, (2012) · Zbl 1319.92057 [32] Arino, J.; Ducrot, A.; Zongo, P., A metapopulation model for malaria with transmission-blocking partial immunity in hosts, J. Math. Biol., 64, 423-448, (2012) · Zbl 1236.92036 [33] Ai, S. B.; Li, J.; Lu, J. L., Mosquito-stage-structured malaria models and their global dynamics, SIAM J. Appl. Math., 72, 1213-1237, (2012) · Zbl 1267.34076 [34] Diekmann, O.; Heesterbeek, J. A.P.; Metz, J. A.J., 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, (1990) · Zbl 0726.92018 [35] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 29-48, (2002) · Zbl 1015.92036 [36] Garba, S. M.; Safi, M. A.; Gumel, A. B., Cross-immunity-induced backward bifurcation for a model of transmission dynamics of two strains of influenza, Nonlinear Anal. RWA, 14, 1384-1403, (2013) · Zbl 1263.92026 [37] Castillo-Chavez, C.; Song, B., Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1, 361-404, (2004) · Zbl 1060.92041 [38] Li, M. Y.; Muldowney, J. S., A geometric approach to global stability problems, SIAM J. Math. Anal., 27, 1070-1083, (1996) · Zbl 0873.34041 [39] Freedman, H. I.; Ruan, S.; Tang, M., Uniform persistence and flows near a closed positively invariant set, J. Dyn. Diff. Equt., 6, 583-600, (1994) · Zbl 0811.34033 [40] Hutson, V.; Schmitt, K., Permanence and the dynamics of biological systems, Math. Biosci., 111, 1-71, (1992) · Zbl 0783.92002 [41] Martin, R. H., Logarithmic norms and projections applied to linear differential systems, J. Math. Anal. Appl., 45, 432-454, (1974) · Zbl 0293.34018 [42] Dhooge, A.; Govaerts, W.; Kuznetsov, Y. A., Matcont: a Matlab package for numerical bifurcation analysis of odes, ACM Trans. Math. Software, 29, 141-164, (2003) · Zbl 1070.65574 [43] Dhooge, A.; Govaerts, W.; Kuznetsov, Y. A.; Meijer, H. G.E.; B. Sautois, New features of the software matcont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst., 14, 145-175, (2008) · Zbl 1158.34302 [44] NHFPC, The situation of malaria in China, http://www.nhfpc.gov.cn/jnr/fznjrgzjz/201404/afcaf6455b264ac5b13c6bf48306b7e9.shtml. [45] Liang, G.-L.; Sun, X.-D.; Wang, J.; Zhang, Z.-X., Sensitivity of plasmodium vivax to chloroquine in laza city, myanmar, Chin. J. Parasitol. Parasit. Dis., 27, 175-176, (2009) [46] Chitnis, N.; Hyman, J. M.; Cushing, J. M., Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70, 1272-1296, (2008) · Zbl 1142.92025 [47] Khan, A.; Hassan, M.; Imran, M., Estimating the basic reproduction number for single-strain dengue fever epidemics, Infect. Dis. Poverty, 12, (2014) [48] Briët, O. J.T., A simple method for calculating mosquito mortality rates, correcting for seasonal variations in recruitment, Med. Vet. Entomol., 16, 22-27, (2002) [49] G. Lu, S. Zhou, O. Horstick, X. Wang, Y. Liu, O. Müller, Malaria outbreaks in China (1990-2013): a systematic review, http://www.malariajournal.com/content/13/1/269. [50] DEDiscover, https://www.dediscover.org/.
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