On a SIR Model in a Patchy Environment Under Constant and Feedback Decentralized Controls with Asymmetric Parameterizations

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.

On Cauchy’s Interlacing Theorem and the Stability of a Class of Linear Discrete Aggregation Models Under Eventual Linear Output Feedback Controls

“…The stability of the equilibrium points of the epidemic models is an interesting topic which is of great relevance to healthcare management. See, for instance, [15][16][17][18][19]. Now, we discuss an epidemic based-model related to stabilization under the given framework of the Cauchy's interlacing theorem.…”

On the Approximated Reachability of a Class of Time-Varying Nonlinear Dynamic Systems Based on Their Linearized Behavior about the Equilibria: Applications to Epidemic Models

“…is the Moore-Penrose pseudoinverse of G c0[0,t * ] (A 0 , b 0 ), which coincides with the inverse if the controllability gramian is non-singular. If G c0[0,t * ] (A 0 , b 0 ) has rank r(≤ n) then it can be factorized as G c0[0,t * ] (A 0 , b 0 ) = G C G D , where G C ∈ R n×r and G D ∈ R r×rn are both of rank r. The subsequent result follows concerning all the set of solutions of (66), or the best approximated solution if (66) is algebraically incompatible, by taking into account the above considerations and the basic related results on pseudoinverse matrices in [15,16]:…”

“…This situation is common in many SEIR models. See, for instance, Reference [15]. Assume that the basic reproduction number [5,6], is less than unity so that the disease-free equilibrium point x e = (S e , 0 , 0 , N(0) − S e ) T 0 is globally asymptotically stable, [3,5,6].…”

“…The usual epidemic models are typically based on differential, difference or mixed equations which describe the coupled dynamics of the various subpopulation or, in general, they can include point and distributed delayed dynamics or to be also formulated in a stochastic framework, [7,8,11]. There are also studies for models of networks available which include different nodes which can represent different sets of interacting communities [14,15], which combined control strategies which take into account the communication links and population flows. Some of the models introduce appropriate either prediction or entropy tools or game theory to discuss the increase of disorder associated to them.…”

“…Section 4 pays also attention to the cases when the controllability gramian is rank-defective, so singular, and either there are non-unique solutions to the point-reachability problem (that is, the system is not controllable and the relevant algebraic system that formulates the problem is compatible indeterminate) or there are no algebraic solutions (that is, such an algebraic system is incompatible). In those cases, the alternative formulation is based on the use of Moore-Penrose pseudoinverses of the controllability gramian [15,16]. Finally, this section also considers the case when the input is saturated, as it is often the case in many real problems, so that the theoretical input to achieve point-reachability of the nominal linearized system has to be "ad hoc" modified.…”