zbMATH — the first resource for mathematics

A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity. (English) Zbl 1117.92039
Summary: We propose a model that tracks the dynamics of malaria in the human host and mosquito vector. Our model incorporates some infected humans that recover from infection and immune humans after loss of immunity to the disease to join the susceptible class again. All the new borne are susceptible to the infection and there is no vertical transmission. The stability of the system is analyzed for the existence of the disease-free and endemic equilibria points.
We established that the disease-free equilibrium point is globally asymptotically stable when the reproduction number, \(R_{0}\leq 1\) and the disease always dies out. For \(R_{0}>1\) the disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable. Thus, due to new births and immunity loss to malaria, the susceptible class will always be refilled and the disease becomes more endemic.

92C60 Medical epidemiology
92C50 Medical applications (general)
34D23 Global stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
Full Text: DOI
[1] Anderson, R.M.; May, R.M., Infectious diseases of humans: dynamics and control, (1991), Oxford University Press Oxford
[2] Aron, J.L.; May, R.M., The population dynamics of malaria, (), 139-179
[3] Aron, J.L., Acquired immunity dependent upon exposure in an SIRS epidemic model, Mathematical biosciences, 88, 37-47, (1988) · Zbl 0637.92007
[4] Aron, J.L., Mathematical modelling of immunity to malaria, Mathematical biosciences, 90, 385-396, (1988) · Zbl 0651.92018
[5] Bailey, N.T.J., The biomathematics of malaria, (1982), Charles Griff London · Zbl 0494.92018
[6] Ghosh, A.K.; Chattopadhyay, J.; Tapaswi, P.K., Immunity boosted by low exposure to infection in an SIRS model, Ecological modelling, 87, 227-233, (1996)
[7] Hale, J.K., Ordinary differential equations, (1969), John Wiley New York · Zbl 0186.40901
[8] Hethcote, H.W., The mathematics of infectious diseases, (2000), SIAM · Zbl 0993.92033
[9] Hethcote, H.W., Qualitative analysis of communicable disease models, Mathematical biosciences, 28, 335-356, (1976) · Zbl 0326.92017
[10] Hirsch, M.W., Systems of differential equations that are competitive or cooperative. V. convergence in 3-dimensional systems, Journal of differential equations, 80, 94-106, (1989) · Zbl 0712.34045
[11] Hirsch, M.W., Systems of differential equations which are competitive or cooperative. IV. structural stability in 3-dimensional systems, SIAM journal on mathematical analysis, 21, 1225-1234, (1990) · Zbl 0734.34042
[12] Hviid, P., Natural acquired immunity to plasmodium falciparum malaria in africa, Acta tropica, 95, 265-269, (2005)
[13] Levins, R.; Awerbuch, T.; Eckardt, U.; Brinkman, I.; Epstein, P.; Makhoul, N.; Albuquerque de Posas, C.; Puccia, C.; Spielman, A.; Wilson, M., The emergence of new diseases, American scientist, 82, 85-118, (1994)
[14] Li, M.Y.; Graef, J.R.; Wang, L.; Karsai, J., Global stability for the SEIR model with varying total population size, Mathematical biosciences, 160, 191-213, (1999) · Zbl 0974.92029
[15] Li, M.Y.; Mouldowney, J.S., Global stability for the SEIR model in epidemiology, Mathematical biosciences, 125, 155-164, (1995) · Zbl 0821.92022
[16] Macdonald, G., The epidemiology and control of malaria, (1957), Oxford university press Oxford
[17] Marsh, K.; Otoo, L.; Hayes, R.H., Antibodies to blood stages antigens of plasmodium falciparum in rural gambians and their relationship to protection against infection, Transactions of the royal society of tropical medicine and hygiene, 83, 293-303, (1989)
[18] McCluskey, C.C.; van den Driessche, P., Global analysis of tuberclosis models, Journal of differential equations, 16, 139-166, (2004) · Zbl 1056.92052
[19] Mouldowney, J.S., Compound matrices and ordinary differential equations, Rocky mountain journal of mathematics, 20, 857-872, (1990) · Zbl 0725.34049
[20] Ngwa, G.A.; Shu, W.S., A mathematical model for endemic malaria with variable human and mosquito populations, Mathematics and computer modelling, 32, 747-763, (2000) · Zbl 0998.92035
[21] Olumese, P., Epidemiology and surveillance: changing the global picture of malaria-myth or reality?, Acta tropica, 95, 265-269, (2005)
[22] Ross, R., The prevention of malaria, (1911), Murry London
[23] Sachs, J.D., A new global effort to control malaria, Science, 298, 122-124, (2002)
[24] Smith, H.L., Monotone dynamical systems. an introduction to the theory of competitive and cooperative systems, (), 231-240
[25] Smith, H.L., Systems of differential equations which generate an order preserving flow, SIAM review, 30, 87-113, (1988) · Zbl 0674.34012
[26] Snow, R.W.; Guerra, C.A.; Noor, A.M.; Myint, H.Y.; Hay, S.I., The global distribution of clinical episodes of plasmodium falciparum, Nature, 434, 214-217, (2005)
[27] Tumwiine, J.; Luboobi, L.S.; Mugisha, J.Y.T., Modelling the effect of treatment and mosquitoes control on malaria transmission, International journal of management and systems, 21, 2, 107-124, (2005)
[28] World Health Organization, World malaria report 2005. Geneva. World Health Organization. WHO/HAM/MAL/2005.1102, 2005.
[29] World Health Organization. Malaria - A global crisis, Geneva, 2000.
[30] Gratz, N.G., Emerging and resulting vector-borne diseases, Annual review of entomology, 44, 51-75, (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.