scholarly journals A Survey on Two Viruses Extensions of Epidemic Model with Continuous and Impulse Control

2021 ◽  
Vol 14 ◽  
pp. 127-154
Author(s):  
Elena Gubar ◽  
◽  
Vladislav Taynitskiy ◽  
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Network Structure ◽  
Epidemic Model ◽  
Impulse Control ◽  
Sir Model ◽  
Initial Population ◽  
Protection Measures ◽  
Virus Spreading ◽  
Theoretical Results

The current study represents a survey on several modifications of compartment epidemic models with continuous and impulse control policies. The main contribution of the survey is the modification of the classical Susceptible Infected Recovered (SIR) model with the assumption that two types of viruses are circulating in the population at the same time. Moreover, we also take into consideration the network structure of the initial population in two-virus SIIR models and estimate the e ectiveness of protection measures over complex networks. In each model, the optimal control problem has been formalized to minimize the costs of the virus spreading and find optimal continuous and impulse antivirus controllers. All theoretical results are corroborated by a large number of numerical simulations.

scholarly journals Optimal Control Problem for Cholera Disease and Cost-Effectiveness Analysis

2021 ◽  
Vol 53 (2) ◽  
pp. 200-2017
Author(s):  
Jhoana Patricia Romero-Leiton ◽  
Muhammad Ozair ◽  
Takasar Hussaing
Keyword(s):  
Optimal Control ◽  
Cost Effectiveness ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Strategies ◽  
Cost Effectiveness Analysis ◽  
Effectiveness Analysis ◽  
Using Data ◽  
Theoretical Results

Cholera is a disease that continues to be a threat to public health globally and is an indicator of inequity and lack of social development in countries. For this reason, strategies for its control need to be investigated. In this work, an optimal control problem related to cholera disease was formulated by introducing personal protection, drug treatment and water sanitation as control strategies. First, the existence and characterization of controls to minimize the performance index or cost function was proved by using classic control theory. Then, the theoretical results were validated with numerical experiments by using data reported in the literature. Finally, the effectiveness and efficiency of the proposed controls were determined through a cost-effectiveness analysis. The results showed that the use of the three controls simultaneously is the cheapest and most effective strategy to control the disease.

scholarly journals An Age and Space Structured SIR Model Describing the Covid-19 Pandemic

2020 ◽  
Author(s):  
Rinaldo M Colombo ◽  
Mauro Garavello ◽  
Francesca Marcellini ◽  
Elena Rossi
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Sir Model ◽  
Qualitative Properties ◽  
Numerical Integrations ◽  
Key Features ◽  
Virus Spreading ◽  
The Basic Reproduction Number

We present an epidemic model capable of describing key features of the present Covid-19 pandemic. While capturing several qualitative properties of the virus spreading, it allows to compute the basic reproduction number, the number of deaths due to the virus and various other statistics. Numerical integrations are used to illustrate the relevance of quarantine and the role of care houses.

An optimal control problem for a spatiotemporal SIR model

2016 ◽  
Vol 6 (1) ◽  
pp. 384-397 ◽  
Author(s):  
Adil El-Alami Laaroussi ◽  
Mostafa Rachik ◽  
Mohamed Elhia
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Sir Model

scholarly journals Optimal Control of Heterogeneous Mutating Viruses

Games ◽  
2018 ◽  
Vol 9 (4) ◽  
pp. 103 ◽  
Author(s):  
Elena Gubar ◽  
Vladislav Taynitskiy ◽  
Quanyan Zhu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Replicator Dynamics ◽  
Optimal Strategies ◽  
Sir Model ◽  
Influenza Viruses ◽  
Human Populations ◽  
Numerical Examples ◽  
Epidemic Season ◽  
Different Strains

Different strains of influenza viruses spread in human populations during every epidemic season. As the size of an infected population increases, the virus can mutate itself and grow in strength. The traditional epidemic SIR model does not capture virus mutations and, hence, the model is not sufficient to study epidemics where the virus mutates at the same time as it spreads. In this work, we establish a novel framework to study the epidemic process with mutations of influenza viruses, which couples the SIR model with replicator dynamics used for describing virus mutations. We formulated an optimal control problem to study the optimal strategies for medical treatment and quarantine decisions. We obtained structural results for the optimal strategies and used numerical examples to corroborate our results.

scholarly journals Vaccination Control in a Stochastic SVIR Epidemic Model

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Peter J. Witbooi ◽  
Grant E. Muller ◽  
Garth J. Van Schalkwyk
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Optimal Control Problem ◽  
Epidemic Model ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Stochastic Problem ◽  
Almost Sure Exponential Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

For a stochastic differential equation SVIR epidemic model with vaccination, we prove almost sure exponential stability of the disease-free equilibrium forR0<1, whereR0denotes the basic reproduction number of the underlying deterministic model. We study an optimal control problem for the stochastic model as well as for the underlying deterministic model. In order to solve the stochastic problem numerically, we use an approximation based on the solution of the deterministic model.

Optimal control of linear systems of arbitrary fractional order

2019 ◽  
Vol 22 (1) ◽  
pp. 170-179 ◽  
Author(s):  
Ivan Matychyn ◽  
Viktoriia Onyshchenko
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Linear Systems ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Support Functions ◽  
Time Optimal ◽  
Control Of Linear Systems ◽  
Theoretical Results

Abstract Optimal control problem for linear systems of arbitrary fractional order in the sense of Riemann–Liouville is treated in the paper. The technique of attainability sets and their support functions is used to obtain sufficient conditions for time-optimal control similar to that of Pontryagin’s maximum principle. Theoretical results are supported by example.

scholarly journals A mathematical investigation of an "SVEIR" epidemic model for the measles transmission

2022 ◽  
Vol 19 (3) ◽  
pp. 2853-2875
Author(s):  
Miled El Hajji ◽  
◽  
Amer Hassan Albargi ◽  
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Global Stability ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Basic Reproduction Number

<abstract><p>A generalized "SVEIR" epidemic model with general nonlinear incidence rate has been proposed as a candidate model for measles virus dynamics. The basic reproduction number $ \mathcal{R} $, an important epidemiologic index, was calculated using the next generation matrix method. The existence and uniqueness of the steady states, namely, disease-free equilibrium ($ \mathcal{E}_0 $) and endemic equilibrium ($ \mathcal{E}_1 $) was studied. Therefore, the local and global stability analysis are carried out. It is proved that $ \mathcal{E}_0 $ is locally asymptotically stable once $ \mathcal{R} $ is less than. However, if $ \mathcal{R} &gt; 1 $ then $ \mathcal{E}_0 $ is unstable. We proved also that $ \mathcal{E}_1 $ is locally asymptotically stable once $ \mathcal{R} &gt; 1 $. The global stability of both equilibrium $ \mathcal{E}_0 $ and $ \mathcal{E}_1 $ is discussed where we proved that $ \mathcal{E}_0 $ is globally asymptotically stable once $ \mathcal{R}\leq 1 $, and $ \mathcal{E}_1 $ is globally asymptotically stable once $ \mathcal{R} &gt; 1 $. The sensitivity analysis of the basic reproduction number $ \mathcal{R} $ with respect to the model parameters is carried out. In a second step, a vaccination strategy related to this model will be considered to optimise the infected and exposed individuals. We formulated a nonlinear optimal control problem and the existence, uniqueness and the characterisation of the optimal solution was discussed. An algorithm inspired from the Gauss-Seidel method was used to resolve the optimal control problem. Some numerical tests was given confirming the obtained theoretical results.</p></abstract>

scholarly journals A Stochastic Optimal Regulator for a Class of Nonlinear Systems

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Gabriella Mavelli ◽  
Giovanni Palombo ◽  
Pasquale Palumbo
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimal Solution ◽  
Bilinear System ◽  
The State ◽  
Affine Transformations ◽  
Average Value ◽  
Stochastic Optimal Control Problem ◽  
Theoretical Results

This work investigates an optimal control problem for a class of stochastic differential bilinear systems, affected by a persistent disturbance provided by a nonlinear stochastic exogenous system (nonlinear drift and multiplicative state noise). The optimal control problem aims at minimizing the average value of a standard quadratic-cost functional on a finite horizon. It has been supposed that neither the state of the system nor the state of the exosystem is directly measurable (incomplete information case). The approach is based on the Carleman embedding, which allows to approximate the nonlinear stochastic exosystem in the form of a bilinear system (linear drift and multiplicative noise) with respect to an extended state that includes the state Kronecker powers up to a chosen degree. This way the stochastic optimal control problem may be restated in a bilinear setting and the optimal solution is provided among all the affine transformations of the measurements. The present work is a nontrivial extension of previous work of the authors, where the Carleman approach was exploited in a framework where only additive noises had been conceived for the state and for the exosystem. Numerical simulations support theoretical results by showing the improvements in the regulator performances by increasing the order of the approximation.

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