Abstract
The paper presents an approach for set-membership estimation of the state of a heterogeneous population in which an infectious disease is spreading. The population state may consist of susceptible, infected, recovered, etc. groups, where the individuals are heterogeneous with respect to traits, relevant to the particular disease. Set-membership estimations in this context are reasonable, since only vague information about the distribution of the population along the space of heterogeneity is available in practice. The presented approach comprises adapted versions of methods which are known in estimation and control theory, and involve solving parametrized families of optimization problems. Since the models of disease spreading in heterogeneous populations involve distributed systems (with non-local dynamics and endogenous boundary conditions), these problems are non-standard. The paper develops the needed theoretical instruments and a solution scheme. SI and SIR models of epidemic diseases are considered as case studies and the results reveal qualitative properties that may be of interest.
Highlights
The role of heterogeneity of a population for the evolution of infectious diseases is well recognized in the existing literature, see e.g. Diekmann et al (1990, 2012), Coutinho et al (1999)
Various kinds of models have been developed to take into account heterogeneity with respect to immune system, contact rates and other traits, including cellular automata (Schneckenreither et al 2006), random networks (Miller 2007; Volz 2008), distributed integro-differential systems (Novozhilov 2008, 2012; Diekmann et al 1990; Veliov 2005), etc
The second goal of the paper is to show that the set-estimation technique may give useful information about the spread of infectious diseases under uncertainty of data
Summary
The role of heterogeneity of a population for the evolution of infectious diseases is well recognized in the existing literature, see e.g. Diekmann et al (1990, 2012), Coutinho et al (1999). In the present paper we employ an alternative approach, in which the distribution of the population among the h-states is uncertain, but set-membership information is available (possibly together with certain aggregated data). The second goal of the paper is to show that the set-estimation technique may give useful information about the spread of infectious diseases under uncertainty of data (we focus on uncertainty of the h-state-distribution of the initial population). We mention that our previous work (Veliov and Widder 2015) allows to determine the exact asymptotics of the aggregated states of a class of heterogeneous SI-models, depending on the initial h-state-distribution of the population This allows to obtain a set-estimation for the asymptotic state of the disease for this particular SI-model in an alternative way. Some technical proofs are given in the Appendices 1 and 2
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