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Diffusion epidemic models with incubation and crisscross dynamics. (English) Zbl 0836.92018
Summary: A diffusion age-structured epidemic model is analyzed. The model describes an epidemic in a host-vector two-population system. Each population is diffusing in a spatial region. Each population is divided into susceptible, incubating, and infectious subclasses. The incubating and infectious subclasses in each population are determined by a structure variable corresponding to age since infection. The model consists of a system of nonlinear partial differential equations with crisscross dynamics. The existence, uniqueness, and asymptotic behavior of solutions are analyzed.

35Q80Applications of PDE in areas other than physics (MSC2000)
35B40Asymptotic behavior of solutions of PDE
35K99Parabolic equations and systems
65M25Method of characteristics (IVP of PDE, numerical methods)
Full Text: DOI
[1] Bailey, N. T. J.: The biomathematics of malaria. (1982) · Zbl 0494.92018
[2] Busenberg, S. N.; Vargas, C.: Modeling chagas’ disease, variable population size and demographic implications. Lect. notes pure appl. Math. 131, 283-295 (1991) · Zbl 0744.92028
[3] Busenberg, S.; Cooke, K.; Iannelli, M.: Endemic thresholds and stability in a class of age-structured epidemics. SIAM J. Appl. math. 48, 1379-1395 (1988) · Zbl 0666.92013
[4] Capasso, V.: Global solution for a diffusive nonlinear deterministic epidemic model. SIAM J. Appl. math. 35, 274-294 (1978) · Zbl 0415.92018
[5] De Mottoni, P.; Orlandi, E.; Tesei, A.: Asymptotic behavior for a system describing epidemics with migration and spatial spread of infection. Nonlin. anal. 3, 663-675 (1979) · Zbl 0416.35009
[6] Diekmann, O.: Limiting behaviour in an epidemic model. Nonlinear anal. 1, No. 5, 459-470 (1977) · Zbl 0371.92024
[7] Diekmann, O.: Thresholds and travelling waves for the geographical spread of infection. J. math. Biol. 6, 109-130 (1978) · Zbl 0415.92020
[8] Dietz, K.: Mathematical models for transmission and control of malaria. Principles and practice of malariology, 1091-1133 (1988)
[9] Fitzgibbon, W. E.; Morgan, J. J.; Waggoner, S. J.: Generalized Lyapunov methods for interactive systems in biology. Lect. notes pure appl. Math. 131, 177-188 (1991) · Zbl 0749.92016
[10] Hoppensteadt, F.: Mathematical theories of populations: demographics, genetics, and epidemics. SIAM reg. Conf. series appl. Math. (1975) · Zbl 0304.92012
[11] Kendall, D. G.: Mathematics and computer science in biology and medicine. 213-225 (1965)
[12] Kubo, M.; Langlais, M.: Periodic solutions for a population dynamics problem with age dependence and spatial structure. J. math. Biol. 29, 363-378 (1991) · Zbl 0718.92018
[13] Langlais, M.: Large time behavior in a non-linear age dependent population dynamics problem with diffusion. J. math. Biol. 26, 319-346 (1988) · Zbl 0713.92019
[14] Langlais, M.: Remarks on an epidemic model with age structure. Lect. notes pure appl. Math. 131, 75-92 (1991) · Zbl 0746.92018
[15] Martin, R. H.: Nonlinear operators and differential equations in Banach spaces. (1976) · Zbl 0333.47023
[16] Murray, J.: Mathematical biology. Biomathematics texts 19 (1989) · Zbl 0682.92001
[17] Nedelman, J.: Introductory review--some new thoughts about some old malaria models. Math. biosci. 73, 159-182 (1985) · Zbl 0567.92020
[18] Olaofe, G.; Olaofe, K.: A simple model for tropical malaria epidemics. Math. biosci. 25, 205-215 (1975) · Zbl 0321.92020
[19] Radcliffe, J.: The effect of the length of incubation period on the velocity of propagation of an epidemic wave. Math. biosci. 19, 257-262 (1974) · Zbl 0311.92017
[20] Ratcliffe, J.; Rass, L.: Wave solutions for deterministic non-reducible n-type epidemics. J. math. Biol. 17, 45-66 (1983) · Zbl 0525.92019
[21] Ratcliffe, J.; Rass, L.: The spatial spread and final size of models for the deterministic host-vector epidemic. Math. biosci. 70, 123-146 (1984) · Zbl 0567.92018
[22] Ratcliffe, J.; Rass, L.: The uniqueness of wave solutions for the deterministic non-reducible n-type epidemic. J. math. Biol. 19, 303-308 (1984) · Zbl 0539.92027
[23] Ratcliffe, J.; Rass, L.: The asymptotic spread of propagation of the deterministic non-reducible n-type epidemic. J. math. Biol. 23, 341-359 (1986) · Zbl 0606.92019
[24] Ratcliffe, J.; Rass, L.: The spatial spread and final size of the deterministic nonreducible n-type epidemic. J. math. Biol. 19, 309-327 (1987)
[25] Thieme, H. R.: A model for the spread of an epidemic. J. math. Biol. 4, 337-351 (1977) · Zbl 0373.92031
[26] Thieme, H. R.: The asymptotic behaviour of solutions of non-linear integral equations. Math. Z. 157, 141-154 (1977) · Zbl 0348.45019
[27] Velasco-Hernandez, J. X.: An epdemiological model for the dynamics of chagas’ disease. Biosystems 26, 127-134 (1991)
[28] Waltman, P.: Deterministic threshold models in the theory of epidemics. (1974) · Zbl 0293.92015
[29] Webb, G. F.: An age-dependent epidemic model with spatial diffusion. Arch. rat. Mech. anal. 75, 91-102 (1980) · Zbl 0484.92018
[30] Webb, G. F.: A reaction-diffusion model for a deterministic diffusive epidemic. J. math. Anal. appl. 84, 150-161 (1981) · Zbl 0484.92019