Abstract
This paper presents a formal description and analysis of an SIR (involving susceptible- infectious-recovered subpopulations) epidemic model in a patchy environment with vaccination controls being constant and proportional to the susceptible subpopulations. The patchy environment is due to the fact that there is a partial interchange of all the subpopulations considered in the model between the various patches what is modelled through the so-called travel matrices. It is assumed that the vaccination controls are administered at each community health centre of a particular patch while either the total information or a partial information of the total subpopulations, including the interchanging ones, is shared by all the set of health centres of the whole environment under study. In the case that not all the information of the subpopulations distributions at other patches are known by the health centre of each particular patch, the feedback vaccination rule would have a decentralized nature. The paper investigates the existence, allocation (depending on the vaccination control gains) and uniqueness of the disease-free equilibrium point as well as the existence of at least a stable endemic equilibrium point. Such a point coincides with the disease-free equilibrium point if the reproduction number is unity. The stability and instability of the disease-free equilibrium point are ensured under the values of the disease reproduction number guaranteeing, respectively, the un-attainability (the reproduction number being less than unity) and stability (the reproduction number being more than unity) of the endemic equilibrium point. The whole set of the potential endemic equilibrium points is characterized and a particular case is also described related to its uniqueness in the case when the patchy model reduces to a unique patch. Vaccination control laws including feedback are proposed which can take into account shared information between the various patches. It is not assumed that there are in the most general case, symmetry-type constrains on the population fluxes between the various patches or in the associated control gains parameterizations.
Highlights
Populations mutually interact through migrations and immigrations to and from other environments
Control is very important in a variety of complex problems where control decisions have to be locally taken for the integrated subsystems due to a lack of full information on the coupling dynamics from and to the remaining coupled subsystems taking part of the whole dynamic systems [10,11,12], the first one concerning with decentralized control while the two last ones are concerned with positivity
The following result, which is proved in Appendix C, relies on the feature that the reproduction number can be reduced by the vaccination controls
Summary
Populations mutually interact through migrations and immigrations to and from other environments. Vaccination strategies are proposed so that each health centre at a particular patch can have and use some certain crossed shared complete or partial information from the remaining patches It is not assumed, in the most general case, that there are symmetry-type constraints related to the mutual interchanges of populations between pairs of patches or in the control gain parameterizations. We point out that patches could be referred to as “nodes” (villages, suburbs, towns or regions, each one with a health centre) while “compartment” is each individual subpopulation of susceptible infectious or recovered at each node and “subsystem” is each SIR epidemic mathematical model located at each node in the sense that its describes the self-dynamics at any patch of the whole model including the effects of couplings to other compartments or subsystems. In our model, the whole system has n subsystems, each one located at one of the n patches, and each subsystem has three compartments, one for each subpopulation
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