The asymptotic behavior of a reducible system of nonlinear integral equations. (English) Zbl 0885.92028
Summary: The methods developed in this paper are motivated mainly by the study of models for rabies. Rabies is a multispecies disease in which the virulence of the virus, and its affect on different species, leads to models where the infection matrix is reducible. The asymptotic behavior of a reducible system of nonlinear integral equations describing the spatio-temporal development of such an epidemic is studied. When the system is nonreducible, an approximate saddle point method can be used for a restricted model with constant infection and removal rates. This approximate method [the authors, ibid. 14, 599-617 (1984; Zbl 0587.92021)] indicated that the asymptotic speed of propagation is \(c_0\), the minimum wave speed. A rigorous analytic proof of this result was given subsequently by the authors in J. Math. Biol. 23, 341-359 (1986; Zbl 0606.92019).
A reducible set of types may be considered as split into nonreducible subsets of types, so that within each subset all types may infect every other type, possibly through a series of infections. For any two subsets, infection in at least one subset cannot cause infection to occur in the other subset. Consider an infection in the \(i\)th subset only, the density of types in the other subsets being taken to be zero. Let \(c_i\) be the corresponding asymptotic speed of propagation. Then, for the full system, the asymptotic speed of propagation differs for the different subsets. Each subset infected will force the epidemic in any subset it infects to propagate with at least its speed of propagation. The approximate saddle point method was again used for the restricted reducible model [the authors, Rocky Mt. J. Math. 23, No. 2, 725-752 (1993; Zbl 0801.92022)]. It indicated that, for a particular subset, the asymptotic speed of propagation is the maximum of the \(c_i\) over all subsets \(i\) which can cause an infection in the particular subset, and which can themselves be infected by the initial infection in the system.
In this paper, with certain conditions imposed, a rigorous proof of these results is obtained for the general reducible model. It is remarkable that these conditions cover not only all cases in which the saddle point method can be applied but also additional cases. A lower bound is also established for the final size of the epidemic for each type, the lower bound holding for all values of the spatial variable.
A reducible set of types may be considered as split into nonreducible subsets of types, so that within each subset all types may infect every other type, possibly through a series of infections. For any two subsets, infection in at least one subset cannot cause infection to occur in the other subset. Consider an infection in the \(i\)th subset only, the density of types in the other subsets being taken to be zero. Let \(c_i\) be the corresponding asymptotic speed of propagation. Then, for the full system, the asymptotic speed of propagation differs for the different subsets. Each subset infected will force the epidemic in any subset it infects to propagate with at least its speed of propagation. The approximate saddle point method was again used for the restricted reducible model [the authors, Rocky Mt. J. Math. 23, No. 2, 725-752 (1993; Zbl 0801.92022)]. It indicated that, for a particular subset, the asymptotic speed of propagation is the maximum of the \(c_i\) over all subsets \(i\) which can cause an infection in the particular subset, and which can themselves be infected by the initial infection in the system.
In this paper, with certain conditions imposed, a rigorous proof of these results is obtained for the general reducible model. It is remarkable that these conditions cover not only all cases in which the saddle point method can be applied but also additional cases. A lower bound is also established for the final size of the epidemic for each type, the lower bound holding for all values of the spatial variable.
MSC:
| 92C60 | Medical epidemiology |
| 45M05 | Asymptotics of solutions to integral equations |
| 45G10 | Other nonlinear integral equations |
Keywords:
reducible systems; \(n\)-type epidemics; spatial spread; pandemic theorem; rabies; speed of propagation; approximate saddle point methodReferences:
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