×

zbMATH — the first resource for mathematics

Four predator prey models with infectious diseases. (English) Zbl 0999.92032
Summary: Four modifications of a predator prey model to include an SIS or SIR parasitic infection are developed and analyzed. Thresholds are identified and global stability results are proved. When the disease persists in the prey population and the predators have a sufficient feeding efficiency to survive, the disease also persists in the predator population.

MSC:
92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
92D40 Ecology
34C60 Qualitative investigation and simulation of ordinary differential equation models
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] ()
[2] ()
[3] Begon, M.; Bowers, R.G., Beyond host-pathogen dynamics, (), 479-509 · Zbl 0839.92024
[4] Anderson, R.M.; May, R.M., The invasion, persistence, and spread of infectious diseases within animal and plant communiites, Phil. trans. R. soc. London, B314, 533-570, (1986)
[5] Hadeler, K.P.; Freedman, H.I., Predator-prey populations with parasitic infection, J. math. biol., 27, 609-631, (1989) · Zbl 0716.92021
[6] Venturino, E., The influence of diseases on Lotka-Volterra systems, Rocky mt. J. math., 24, 381-402, (1994) · Zbl 0799.92017
[7] Venturino, E., Epidemics in predator-prey models: disease in the prey, (), 381-393
[8] Hudson, P.J.; Dobson, A.P.; Newborn, D., Do parasites make prey more vulnerable to predation? red grouse and parasites, J. anim. ecol., 61, 681-692, (1992)
[9] Chattopadhyay, J.; Arino, O., A predator-prey model with disease in the prey, Nonlinear anal., 36, 747-766, (1999) · Zbl 0922.34036
[10] H.W. Hethcote, The mathematics of infectious diseases, SIAM (to appear). · Zbl 0993.92033
[11] De Jong, M.C.M.; Diekmann, O.; Heesterbeek, J.A.P., How does transmission depend on population size?, (), 84-94 · Zbl 0850.92042
[12] Gao, L.Q.; Hethcote, H.W., Disease transmission models with density-dependent demographics, J. math. biol., 30, 717-731, (1992) · Zbl 0774.92018
[13] Hethcote, H.W., Qualitative analyses of communicable disease models, Math. biosci., 28, 335-356, (1976) · Zbl 0326.92017
[14] Hethcote, H.W.; Van Ark, J.W., Epidemiological models with heterogeneous populations: proportionate mixing, parameter estimation and immunization programs, Math. biosci., 84, 85-118, (1987) · Zbl 0619.92006
[15] Mena-Lorca, J.; Hethcote, H.W., Dynamic models of infectious diseases as regulators of population sizes, J. math. biol., 30, 693-716, (1992) · Zbl 0748.92012
[16] Hethcote, H.W.; Stech, H.W.; van den Driessche, P., Periodicity and stability in epidemic models: A survey, (), 65-82
[17] Pielou, E.C., Introduction to mathematical ecology, (1969), Wiley-Interscience New York · Zbl 0259.92001
[18] Nisbet, R.M.; Gurney, W.S.C., Modelling fluctuating populations, (1982), Wiley-Interscience New York · Zbl 0593.92013
[19] Thieme, H.R., Convergence results and a poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. math. biol., 30, 755-763, (1992) · Zbl 0761.34039
[20] Miller, R.K.; Michel, A.N., Ordinary differential equations, (1982), Academic Press New York · Zbl 0499.34024
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.