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

A comprehensive analysis of COVID-19 nonlinear mathematical model by incorporating the environment and social distancing

This research conducts a detailed analysis of a nonlinear mathematical model representing COVID-19, incorporating both environmental factors and social distancing measures. It thoroughly analyzes the model's equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. The study develops a sophisticated stability theory, primarily focusing on the characteristics of the Volterra-Lyapunov (V-L) matrices method. To understand the dynamic behavior of COVID-19, numerical simulations are essential. For this purpose, the study employs a robust numerical technique known as the non-standard finite difference (NSFD) method, introduced by Mickens. Various results are visually presented through graphical representations across different parameter values to illustrate the impact of environmental factors and social distancing measures.

Dynamics analysis of an SVEIR epidemic model in a patchy environment

In this paper, we propose a multi-patch SVEIR epidemic model that incorporates vaccination of both newborns and susceptible populations. We determine the basic reproduction number $ R_{0} $ and prove that the disease-free equilibrium $ P_{0} $ is locally and globally asymptotically stable if $ R_{0} < 1, $ and it is unstable if $ R_{0} > 1. $ Moreover, we show that the disease is uniformly persistent in the population when $ R_{0} > 1. $ Numerical simulations indicate that vaccination strategies can effectively control disease spread in all patches while population migration can either intensify or prevent disease transmission within a patch.

Optimal Control and Stability Analysis of an SEIR Model with Infectious Force in Latent Period

In this paper, an SEWIR epidemic model with the government control rate and infectious force in latent period is proposed. The conditions to the existence and uniqueness of disease-free and endemic equilibrium points in the SEWIR model are obtained. By using the Hurwitz criterion, the locally asymptotic stability of disease-free and endemic equilibrium points is proved. We show the global asymptotic stability of the disease-free equilibrium point by the construction of Lyapunov function and LaSalle invariance principle. The globally asymptotic stability of the endemic equilibrium is verified by numerical simulation. Several optimal control strategies are proposed on controlling infectious diseases.

The Global dynamics of a SIR model considering competitions among multiple strains in patchy environments

Pandemics, such as Covid-19 and AIDS, tend to be highly contagious and have the characteristics of global spread and existence of multiple virus strains. To analyze the competition among different strains, a high dimensional SIR model studying multiple strains' competition in patchy environments is introduced in this work. By introducing the basic reproductive number of different strains, we found global stability conditions of disease-free equilibrium and persistence conditions of the model. The competition exclusion conditions of that model are also given. This work gives some insights into the properties of the multiple strain patchy model and all of the analysis methods used in this work could be used in other related high dimension systems.

On the Supervision of a Saturated SIR Epidemic Model with Four Joint Control Actions for a Drastic Reduction in the Infection and the Susceptibility through Time

This paper presents and studies a new epidemic SIR (Susceptible–Infectious–Recovered) model with susceptible recruitment and eventual joint vaccination efforts for both newborn and susceptible individuals. Furthermore, saturation effects in the infection incidence terms are eventually assumed for both the infectious and the susceptible subpopulations. The vaccination action on newborn individuals is assumed to be applied to a fraction of them while that on the susceptible general population is of linear feedback type reinforced with impulsive vaccination actions (in practice, very strong and massive vaccination controls) at certain time points, based on information on the current levels of the susceptible subpopulation. Apart from the above vaccination controls, it is also assumed that the average of contagion contacts can be controlled via intervention measures, such as confinements or isolation measures, social distance rules, use of masks, mobility constraints, etc. The main objectives of the paper are the achievement of a strictly decreasing infection for all time periods and that of the susceptible individuals over the initial period if they exceed the disease-free equilibrium value. The monitoring mechanism is the combined activation of intervention measures to reduce the contagion contacts together with the impulsive vaccination to reduce susceptibility. The susceptibility and recovery levels of the disease-free equilibrium point are suitably prefixed by the design of the regular feedback vaccination on the susceptible subpopulation.

Design a robust sliding mode controller based on the state and parameter estimation for the nonlinear epidemiological model of Covid-19

In this research, the challenging problem of Covid-19 mitigation is looked at from an engineering point of view. At first, the behavior of coronavirus in the Iranian and Russian societies is expressed by a set of ordinary differential equations. In the proposed model, the control input signals are vaccination, social distance and facial masks, and medical treatment. The unknown parameters of the system are estimated by long short-term memory (LSTM) algorithm. In the LSTM algorithm, the problem of long-term dependency is prevented. The uncertainty and measurement noises are inherent characteristics of epidemiological models. For this reason, an extended Kalman filter (EKF) is developed to estimate the state variables of the proposed model. In continuation, a robust sliding mode controller is designed to control the spread of coronavirus under vaccination, social distance and facial masks, and medical treatment. The stability of the closed-loop system is guaranteed by the Lyapunov theorems. The official confirmed data provided by the Iranian and Russian ministries of health are employed to simulate the proposed algorithms. It is understood from simulation results that global vaccination has the potential to create herd immunity in long term. Under the proposed controller, daily Covid-19 infections and deaths become less than 500 and 10 people, respectively.

A Study on COVID-19 Incidence in Europe through Two SEIR Epidemic Models Which Consider Mixed Contagions from Asymptomatic and Symptomatic Individuals

The impact of the SARS-CoV-2 (COVID-19) on the world has been partially controlled through different measures of social isolation and prophylaxis. Two new SEIR (Susceptible-Exposed-Infected-Recovered) models are proposed in order to describe this spread through different countries of Europe. In both models the infectivity of the asymptomatic period during the exposed stage of the disease will be taken into account. The different transmission rates of the SEIR models are calculated by considering the different locations and, more importantly, the lockdown measures implemented in each region. A new classification of these intervention measures will be set and their influence on the values of the transmission rates will be estimated through regression analysis.

A Modelization of the Propagation of COVID-19 in Regions of Spain and Italy with Evaluation of the Transmission Rates Related to the Intervention Measures

Two discrete mathematical SIR models (Susceptible-Infectious-Recovered) are proposed for modelling the propagation of the SARS-CoV-2 (COVID-19) through Spain and Italy. One of the proposed models is delay-free while the other one considers a delay in the propagation of the infection. The objective is to estimate the transmission, also known as infectivity rate, through time taking into account the infection evolution data supplied by the official health care systems in both countries. Such a parameter is estimated through time at different regional levels and it is seen to be strongly dependent on the intervention measures such as the total (except essential activities) or partial levels of lockdown. Typically, the infectivity rate evolves towards a minimum value under total lockdown and it increases again when the confinement measures are partially or totally removed.

An Approach for the Global Stability of Mathematical Model of an Infectious Disease

The global stability analysis for the mathematical model of an infectious disease is discussed here. The endemic equilibrium is shown to be globally stable by using a modification of the Volterra–Lyapunov matrix method. The basis of the method is the combination of Lyapunov functions and the Volterra–Lyapunov matrices. By reducing the dimensions of the matrices and under some conditions, we can easily show the global stability of the endemic equilibrium. To prove the stability based on Volterra–Lyapunov matrices, we use matrices with the symmetry properties (symmetric positive definite). The results developed in this paper can be applied in more complex systems with nonlinear incidence rates. Numerical simulations are presented to illustrate the analytical results.

On Confinement and Quarantine Concerns on an SEIAR Epidemic Model with Simulated Parameterizations for the COVID-19 Pandemic

This paper firstly studies an SIR (susceptible-infectious-recovered) epidemic model without demography and with no disease mortality under both total and under partial quarantine of the susceptible subpopulation or of both the susceptible and the infectious ones in order to satisfy the hospital availability requirements on bed disposal and other necessary treatment means for the seriously infectious subpopulations. The seriously infectious individuals are assumed to be a part of the total infectious being described by a time-varying proportional function. A time-varying upper-bound of those seriously infected individuals has to be satisfied as objective by either a total confinement or partial quarantine intervention of the susceptible subpopulation. Afterwards, a new extended SEIR (susceptible-exposed-infectious-recovered) epidemic model, which is referred to as an SEIAR (susceptible-exposed-symptomatic infectious-asymptomatic infectious-recovered) epidemic model with demography and disease mortality is given and focused on so as to extend the above developed ideas on the SIR model. A proportionally gain in the model parameterization is assumed to distribute the transition from the exposed to the infectious into the two infectious individuals (namely, symptomatic and asymptomatic individuals). Such a model is evaluated under total or partial quarantines of all or of some of the subpopulations which have the effect of decreasing the number of contagions. Simulated numerical examples are also discussed related to model parameterizations of usefulness related to the current COVID-19 pandemic outbreaks.

On the Use of Entropy Issues to Evaluate and Control the Transients in Some Epidemic Models

This paper studies the representation of a general epidemic model by means of a first-order differential equation with a time-varying log-normal type coefficient. Then the generalization of the first-order differential system to epidemic models with more subpopulations is focused on by introducing the inter-subpopulations dynamics couplings and the control interventions information through the mentioned time-varying coefficient which drives the basic differential equation model. It is considered a relevant tool the control intervention of the infection along its transient to fight more efficiently against a potential initial exploding transmission. The study is based on the fact that the disease-free and endemic equilibrium points and their stability properties depend on the concrete parameterization while they admit a certain design monitoring by the choice of the control and treatment gains and the use of feedback information in the corresponding control interventions. Therefore, special attention is paid to the evolution transients of the infection curve, rather than to the equilibrium points, in terms of the time instants of its first relative maximum towards its previous inflection time instant. Such relevant time instants are evaluated via the calculation of an “ad hoc” Shannon’s entropy. Analytical and numerical examples are included in the study in order to evaluate the study and its conclusions.

On the Entropy of Events under Eventually Global Inflated or Deflated Probability Constraints. Application to the Supervision of Epidemic Models under Vaccination Controls

This paper extends the formulation of the Shannon entropy under probabilistic uncertainties which are basically established in terms or relative errors related to the theoretical nominal set of events. Those uncertainties can eventually translate into globally inflated or deflated probabilistic constraints. In the first case, the global probability of all the events exceeds unity while in the second one lies below unity. A simple interpretation is that the whole set of events losses completeness and that some events of negative probability might be incorporated to keep the completeness of an extended set of events. The proposed formalism is flexible enough to evaluate the need to introduce compensatory probability events or not depending on each particular application. In particular, such a design flexibility is emphasized through an application which is given related to epidemic models under vaccination and treatment controls. Switching rules are proposed to choose through time the active model, among a predefined set of models organized in a parallel structure, which better describes the registered epidemic evolution data. The supervisory monitoring is performed in the sense that the tested accumulated entropy of the absolute error of the model versus the observed data is minimized at each supervision time-interval occurring in-between each two consecutive switching time instants. The active model generates the (vaccination/treatment) controls to be injected to the monitored population. In this application, it is not proposed to introduce a compensatory event to complete the global probability to unity but instead, the estimated probabilities are re-adjusted to design the control gains.

Locating Multiple Sources of Contagion in Complex Networks under the SIR Model

Simultaneous outbreaks of contagion are a great threat against human life, resulting in great panic in society. It is urgent for us to find an efficient multiple sources localization method with the aim of studying its pathogenic mechanism and minimizing its harm. However, our ability to locate multiple sources is strictly limited by incomplete information about nodes and the inescapable randomness of the propagation process. In this paper, we present a valid approach, namely the Potential Concentration Label method, which helps locate multiple sources of contagion faster and more accurately in complex networks under the SIR(Susceptible-Infected-Recovered) model. Through label assignment in each node, our aim is to find the nodes with maximal value after several iterations. The experiments demonstrate that the accuracy of our multiple sources localization method is high enough. With the number of sources increasing, the accuracy of our method declines gradually. However, the accuracy remains at a slight fluctuation when average degree and network scale make a change. Moreover, our method still keeps a high multiple sources localization accuracy with noise of various intensities, which shows its strong anti-noise ability. I believe that our method provides a new perspective for accurate and fast multi-sources localization in complex networks.

On the Design of Hyperstable Feedback Controllers for a Class of Parameterized Nonlinearities. Two Application Examples for Controlling Epidemic Models

This paper studies the hyperstability and the asymptotic hyperstability of a single-input single-output controlled dynamic system whose feed-forward input-output dynamics is nonlinear and eventually time-varying consisting of a linear nominal part, a linear incremental perturbed part and a nonlinear and eventually time-varying one. The nominal linear part is described by a positive real transfer function while the linear perturbation is defined by a stable transfer function. The nonlinear and time-varying disturbance is, in general, unstructured but it is upper-bounded by the combination of three additive absolute terms depending on the input, output and input-output product, respectively. The non-linear time-varying feedback controller is any member belonging to a general class which satisfies an integral Popov's-type inequality. This problem statement allows the study of the conditions guaranteeing the robust stability properties under a variety of the controllers designed for the controlled system and controller disturbances. In this way, set of robust hyperstability and asymptotic hyperstability of the closed-loop system are given based on the fact that the input-output energy of the feed-forward controlled system is positive and bounded for all time and any given initial conditions and controls satisfying Popov's inequality. The importance of those hyperstability and asymptotic hyperstability properties rely on the fact that they are related to global closed-loop stability, or respectively, global closed-loop asymptotic stability of the same uncontrolled feed-forward dynamics subject to a great number of controllers under the only condition that that they satisfy such a Popov's-type inequality. It is well-known the relevance of vaccination and treatment controls for Public Health Management at the levels of prevention and healing. Therefore, two application examples concerning the linearization of known epidemic models and their appropriate vaccination and/or treatment controls on the susceptible and infectious, respectively, are discussed in detail with the main objective in mind of being able of achieving a fast convergence of the state- trajectory solutions to the disease- free equilibrium points under a wide class of control laws under deviations of the equilibrium amounts of such populations.

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

This paper links the celebrated Cauchy’s interlacing theorem of eigenvalues for partitioned updated sequences of Hermitian matrices with stability and convergence problems and results of related sequences of matrices. The results are also applied to sequences of factorizations of semidefinite matrices with their complex conjugates ones to obtain sufficiency-type stability results for the factors in those factorizations. Some extensions are given for parallel characterizations of convergent sequences of matrices. In both cases, the updated information has a Hermitian structure, in particular, a symmetric structure occurs if the involved vector and matrices are complex. These results rely on the relation of stable matrices and convergent matrices (those ones being intuitively stable in a discrete context). An epidemic model involving a clustering structure is discussed in light of the given results. Finally, an application is given for a discrete-time aggregation dynamic system where an aggregated subsystem is incorporated into the whole system at each iteration step. The whole aggregation system and the sequence of aggregated subsystems are assumed to be controlled via linear-output feedback. The characterization of the aggregation dynamic system linked to the updating dynamics through the iteration procedure implies that such a system is, generally, time-varying.