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Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. (English) Zbl 1015.92036
Summary: A precise definition of the basic reproduction number, \(\mathcal R_0\), is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if \(\mathcal R_0<1\) , then the disease free equilibrium is locally asymptotically stable; whereas if \(\mathcal R_0>1\), then it is unstable. Thus, \({\mathcal R}_0\) is a threshold parameter for the model.
An analysis of the local centre manifold yields a simple criterion for the existence and stability of super- and sub-threshold endemic equilibria for \(\mathcal R_0\) near one. This criterion, together with the definition of \(\mathcal R_0\), is illustrated by treatment, multigroup, staged progression, multistrain and vector-host models and can be applied to more complex models. The results are significant for disease control.

MSC:
92D30 Epidemiology
34C60 Qualitative investigation and simulation of ordinary differential equation models
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[1] Hethcote, H.W., The mathematics of infectious diseases, SIAM rev., 42, 599, (2000) · Zbl 0993.92033
[2] Diekmann, O.; Heesterbeek, J.A.P.; Metz, J.A.J., On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. math. biol., 28, 365, (1990) · Zbl 0726.92018
[3] Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos, (1990), Springer Berlin · Zbl 0701.58001
[4] Berman, A.; Plemmons, R.J., Nonnegative matrices in the mathematical sciences, (1970), Academic Press New York
[5] Anderson, R.M.; May, R.M., Infectious diseases of humans, (1991), Oxford University Oxford
[6] Diekmann, O.; Heesterbeek, J.A.P., Mathematical epidemiology of infectious diseases: model building, analyis and interpretation, (1999), Wiley New York · Zbl 0997.92505
[7] Hethcote, H.W.; Ark, J.W.V., Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs, Math. biosci., 84, 85, (1987) · Zbl 0619.92006
[8] Huang, W.; Cooke, K.L.; Castillo-Chavez, C., Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. appl. math., 52, 3, 835, (1992) · Zbl 0769.92023
[9] Castillo-Chavez, C.; Feng, Z.; Huang, W., On the computation of R0 and its role in global stability, (), 229
[10] Nold, A., Heterogeneity in disease-transmission modeling, Math. biosci., 52, 227, (1980) · Zbl 0454.92020
[11] Hyman, J.M.; Li, J.; Stanley, E.A., The differential infectivity and staged progression models for the transmission of HIV, Math. biosci., 155, 77, (1999) · Zbl 0942.92030
[12] Castillo-Chavez, C.; Feng, Z., To treat of not to treat: the case of tuberculosis, J. math. biol., 35, 629, (1997) · Zbl 0895.92024
[13] Blower, S.M.; Small, P.M.; Hopewell, P.C., Control strategies for tuberculosis epidemics: new models for old problems, Science, 273, 497, (1996)
[14] Hethcote, H.W., An immunization model for a heterogeneous population, Theor. populat. biol., 14, 338, (1978) · Zbl 0392.92009
[15] Dushoff, J.; Huang, W.; Castillo-Chavez, C., Backwards bifurcations and catastrophe in simple models of fatal diseases, J. math. biol., 36, 227, (1998) · Zbl 0917.92022
[16] Jacquez, J.A.; Simon, C.P.; Koopman, J.; Sattenspiel, L.; Perry, T., Modelling and analyzing HIV transmission: the effect of contact patterns, Math. biosci., 92, 119, (1988) · Zbl 0686.92016
[17] Feng, Z.; Velasco-Hernández, J.X., Competitive exclusion in a vector – host model for the dengue fever, J. math. biol., 35, 523, (1997) · Zbl 0878.92025
[18] Castillo-Chavez, C.; Feng, Z.; Capurro, A.F., A model for TB with exogenous reinfection, Theor. populat. biol., 57, 235, (2000) · Zbl 0972.92016
[19] D. Greenhalgh, O. Diekmann, M.C.M. de Jong, Subcritical endemic steady states in mathematical models for animal infections with incomplete immunity, Math. Biosci. 165 (2000) 1 · Zbl 0983.92007
[20] Busenberg, S.; van den Driessche, P., Disease transmission in multigroup populations of variable size, (), 15
[21] Lin, X.; Hethcote, H.W.; van den Driessche, P., An epidemiological model for HIV/AIDS with proportional recruitment, Math. biosci., 118, 181, (1993) · Zbl 0793.92011
[22] Horn, R.A.; Johnson, C.R., Topics in matrix analysis, (1991), Cambridge University Cambridge · Zbl 0729.15001
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