## A mathematical model for endemic malaria with variable human and mosquito populations.(English)Zbl 0998.92035

From the paper: We develop and analyse a model that incorporates compartments for mosquito populations. Following the ideas advanced by J.L. Aron [Math. Biosci. 64, 249-259 (1983; Zbl 0515.92026); ibid. 88, No. 1, 37-47 (1988; Zbl 0637.92007); ibid. 90, No. 1/2, 385-396 (1988; Zbl 0651.92018)], we introduce in our model a class of persons who are partially immune to the disease malaria, but who may be infectious. We assume density dependent death rates in both vector and human populations so that the total populations are varying with time through a modification of the logistic equation that includes disease related deaths.
A deterministic differential equation model for endemic malaria involving variable human and mosquito populations is analysed. Conditions are derived for the existence of endemic and disease-free equilibria. A threshold parameter $$\widetilde{R}_0$$ exists and the disease can persist if and only if $$\widetilde{R}_0$$ exceeds 1. The disease-free equilibrium always exists and is globally stable when $$\widetilde{R}_0$$ is below 1. Numerical simulations show that the endemic equilibrium, when it exists, is unique and globally stable.
The paper is organised as follows. In Section 2, we briefly outline the derivation of the model and investigate the existence of steady states in Section 3. A linear stability analysis around the steady states is performed in Section 4, where we show that the disease-free equilibrium is globally stable. In Section 5, we present some numerical simulations, and round up the paper with some concluding remarks in Section 6.

### MSC:

 92D30 Epidemiology 92C60 Medical epidemiology 34C60 Qualitative investigation and simulation of ordinary differential equation models 34D23 Global stability of solutions to ordinary differential equations

### Citations:

Zbl 0515.92026; Zbl 0637.92007; Zbl 0651.92018
Full Text:

### References:

 [1] Bradley, T; Department of Microbiology Immunology of Liecester, Malaria and drug resistance, (1996), http://www-micro.msb.le.ac.uk/224/Bradley/Bradley.html [2] Giles, H.M; Warrel, D.A, Bruce-Chwatt’s essential malariology, (1993), Heinemann Medical Books [3] WHO, Selected health problems, World health statistics quarterly, 44, 189-197, (1991) [4] WHO, Report: division of control of tropical diseases, (1999), latest update [5] WHO, The urban crises, World health statistics quarterly, 44, 208-233, (1991) [6] WHO, The role and participation of European countries in the fight against malaria, () [7] Elizabeth, A.C; Newton; Reiter, P, A model for the transmission of dengue fever with an evaluation of the impact of ultra-low volume (ulv) insecticide applications on dengue epidemics, Am. J. trop. med. hyg., 47, 6, 709-720, (1992) [8] Rossignol, P.A; Ribiero, J.M.C; Spielman, A, Increased biting rate and reduced fertility in sporozoite infected mosquitos, Am. J. trop. med. hyg., 35, 277-279, (1986) [9] Hethcote, H.W; Stech, H.W; van den Driessche, P, Periodicity and stability in epidemic models: A survey, (), 65-82 [10] Ross, R, The prevention of malaria, (1911), John Murray London [11] Macdonald, G, The analysis of infection rates in diseases in which superinfection occurs, Trop. dis. bull., 47, 907-915, (1950) [12] Macdonald, G, The analysis of sporozoite rate, Trop. dis. bull., 49, 569-585, (1952) [13] Macdonald, G, The epidemiology and control of malaria, (1957), Oxford University Press London [14] Molineaux, L; Gramiccia, G, The garki project, research on the epidemiology and control of malaria in the sudan savana of west africa, (1980), WHO Geneva [15] Dietz, K; Molineaux, L; Thomas, A, A malaria model tested in the african savana, Bull. WHO, 50, 347-357, (1974) [16] Nedelman, J, Inoculation and recovery rates in the malaria model of dietz, molineaux, and Thomas, Math. biosci., 69, 209-233, (1983) · Zbl 0529.92019 [17] Singer, B; Cohen, J.E, Estimating malaria incidence and recovery rates from panel surveys, Math. biosci., 49, 273-305, (1980) · Zbl 0425.92011 [18] Gatton, M; Hogarth, W; Saul, A; Dayananda, P, A model for predicting the transmission rate of malaria from serological data, J. math. biol., 34, 878-888, (1996) · Zbl 0858.92021 [19] Aron, J.L; May, R.M, The population dynamics of malaria, () [20] Nedelman, J, Introductory review: some new thoughts about some old malaria models, Math. biosci., 73, 158-182, (1985) · Zbl 0567.92020 [21] Benenson, A.S, Control of communicable diseases in man, (1990), American Public Health Association Washington, DC [22] Anderson, R.M; May, R.M, Population biology of infectious diseases, Nature, 280, 361-367, (1979), Part 1 [23] Anderson, R.M; May, R.M, Infectious diseases of humans: dynamics and control, (1991), Oxford University Press Oxford [24] Aron, J.L, Dynamics of acquired immunity boosted by exposure to infection, Math. biosci., 64, 249-253, (1983) · Zbl 0515.92026 [25] Aron, J.L, Mathematical modelling of immunity to malaria, Math. biosci., 90, 385-396, (1988) · Zbl 0651.92018 [26] Aron, J.L, Acquired immunity dependent upon exposure in an SIRS epidemic model, Math. biosci., 88, 37-47, (1988) · Zbl 0637.92007 [27] Gao, L.Q; Hethcote, H.W, Disease transmission models with density-dependent demographics, J. math. biol., 30, 693-716, (1992) · Zbl 0774.92018 [28] Verhulst, P.P, Notice sur la loi que la population suit dans son acroissement, Correspondence mathématique et physique, 10, 113-121, (1838) [29] Nisbet, R.M; Gurney, W.S.C, Modelling fluctuating populations, (1982), John Wiley and Sons Chichester · Zbl 0593.92013 [30] Shears, P, Epidemiology and infection in famine and disasters, Epidemiol. infect., 107, 241-251, (1991) [31] Matessi, C, A theoretical approach to the dynamics of single populations, (), 223-247 · Zbl 0452.92021 [32] Greenhalgh, D, Hopf bifurcation in epidemic models with latent period and nonpermanent immunity, Mathl. comput. modelling, 25, 2, 85-107, (1997) · Zbl 0877.92023 [33] Esteva, L; Vargas, C, A model for the dengue disease with a variable human population, J. math. biol., 38, 220-240, (1999) · Zbl 0981.92016 [34] Nåsell, I, Hybrid models of tropical infections, (), 51-107 [35] Anderson, R.M, Mathematical and statistical study of the epidemiology of HIV, J. aids, 3, 333-346, (1989) [36] Thieme, H.R, Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. biosci., 111, 99-130, (1992) · Zbl 0782.92018 [37] Hale, J.K, Ordinary differential equations, (), 296-297 · Zbl 0186.40901 [38] Busenberg, S; van den Driessche, P, Analysis of a disease transmission model in a population with varying size, J. math. biol., 28, 257-270, (1990) · Zbl 0725.92021 [39] Busenberg, S; Cooke, K.L; Thieme, H, Demographic change and persistence of HIV/AIDS in heterogeneous population, (1989), (Preprint) [40] Diekmann, O; Heesterbeek, J.A.P; Metz, J.A.J, On the definition and computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. math. biol., 28, 365-382, (1990) · Zbl 0726.92018 [41] Hahn, W, Stability of motion, (1967), Springer-Verlag Berlin, Translation by P. Arne Baartz · Zbl 0189.38503 [42] Dietz, K, Models for parasitic disease control, Bull. intern. statist. inst., 46, 531-544, (1975) [43] Dietz, K, On the transmission dynamics of HIV, Math. biosci., 90, 397-414, (1988) · Zbl 0651.92019 [44] Esteva, L; Vargas, C, Analysis of a dengue disease transmission model, Math. biosci., 150, 131-151, (1998) · Zbl 0930.92020
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.