On the effect of population heterogeneity on dynamics of epidemic diseases
Tóm tắt
The paper investigates a class of SIS models of the evolution of an infectious disease in a heterogeneous population. The heterogeneity reflects individual differences in the susceptibility or in the contact rates and leads to a distributed parameter system, requiring therefore, distributed initial data, which are often not available. It is shown that there exists a corresponding homogeneous (ODE) population model that gives the same aggregated results as the distributed one, at least in the expansion phase of the disease. However, this ODE model involves a nonlinear “prevalence-to-incidence” function which is not constructively defined. Based on several established properties of this function, a simple class of approximating function is proposed, depending on three free parameters that could be estimated from scarce data. How the behaviour of a population depends on the level of heterogeneity (all other parameters kept equal) – this is the second issue studied in the paper. It turns out that both for the short run and for the long run behaviour there exist threshold values, such that more heterogeneity is advantageous for the population if and only if the initial (weighted) prevalence is above the threshold.
Tài liệu tham khảo
Coutinho, F.A.B., Massad, E., Lopez, L.F., Burattini, M.N.: Modelling Heterogeneity in individual frailties in epidemic models. Math. Comput. Modelling 30, 97–115 (1999)
Diekmann, O., Heesterbeek, J.A.P.: Mathematical epidemiology of infectious diseases. Model building, analysis and interpretation. Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2000
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, 365–382 (1990)
Feichtinger, G., Tsachev, Ts., Veliov, V.M.: Maximum principle for age and duration structured systems: a tool for optimal prevention and treatment of HIV. Math. Population Studies 11 (1), 3–28 (2004)
Gavrila, C., Pollack, H.A., Caulkins, J.P., Kort, P.M., Feichtinger, G., Tragler, G.: Optimal control of harm reduction in preventing blood-borne diseases among drug users, In submission
Murray, J.D.: Math. Biol. Springer, 1989
Sanderson, W.C.: The demographic impact of HIV medication programs: with examples from Botswana. Paper presented at the Population Association of America Meetings, Atlanta, GA, May, 2002
Thieme, H.R., Castillo-Chavez C.: How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS? SIAM J. Appl. Math. 53, 1447–1479 (1993)
Veliov, V.M.: Newton’s method for problems of optimal control of heterogeneous systems. Optimization Methods and Software 18 (6), 689–703 (2003)