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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, $\cal R_0$, is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if $\cal R_0<1$ , then the disease free equilibrium is locally asymptotically stable; whereas if $\cal R_0>1$, then it is unstable. Thus, ${\cal R}_0$ 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 $\cal R_0$ near one. This criterion, together with the definition of $\cal R_0$, 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.

34C60Qualitative investigation and simulation of models (ODE)
Full Text: DOI
[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) · Zbl 0701.58001
[4] Berman, A.; Plemmons, R. J.: Nonnegative matrices in the mathematical sciences. (1970) · Zbl 0484.15016
[5] Anderson, R. M.; May, R. M.: Infectious diseases of humans. (1991)
[6] Diekmann, O.; Heesterbeek, J. A. P.: Mathematical epidemiology of infectious diseases: model building, analyis and interpretation. (1999) · 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, No. 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. Mathematical approaches for emerging and reemerging infectious diseases: an introduction, 229 (2002)
[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.; Den Driessche, P. Van: Disease transmission in multigroup populations of variable size. Theory of epidemics 1, 15 (1995)
[21] Lin, X.; Hethcote, H. W.; Den Driessche, P. Van: 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) · Zbl 0729.15001