Optimal Strategies for Control of COVID-19: A Mathematical Perspective

Disease Free ◽

A deterministic ordinary differential equation model for SARS-CoV-2 is developed and analysed, taking into account the role of exposed, mildly symptomatic, and severely symptomatic persons in the spread of the disease. It is shown that in the absence of infective immigrants, the model has a locally asymptotically stable disease-free equilibrium whenever the basic reproduction number is below unity. In the absence of immigration of infective persons, the disease can be eradicated whenever ℛ 0 < 1 . Specifically, if the controls u i , i = 1,2,3,4 , are implemented to 100% efficiency, the disease dies away easily. It is shown that border closure (or at least screening) is indispensable in the fight against the spread of SARS-CoV-2. Simulation of optimal control of the model suggests that the most cost-effective strategy to combat SARS-CoV-2 is to reduce contact through use of nose masks and physical distancing.

A Stochastic TB Model for a Crowded Environment

Journal of Applied Mathematics ◽

10.1155/2018/3420528 ◽

2018 ◽

Vol 2018 ◽

pp. 1-8 ◽

Cited By ~ 3

Author(s):

Sibaliwe Maku Vyambwera ◽

Peter Witbooi

Keyword(s):

Compartmental Model ◽

Deterministic Model ◽

Reproduction Number ◽

Stochastic Perturbation ◽

Equation Model ◽

Ordinary Differential Equation Model ◽

The Stability ◽

Crowded Environments ◽

Disease Free

We propose a stochastic compartmental model for the population dynamics of tuberculosis. The model is applicable to crowded environments such as for people in high density camps or in prisons. We start off with a known ordinary differential equation model, and we impose stochastic perturbation. We prove the existence and uniqueness of positive solutions of a stochastic model. We introduce an invariant generalizing the basic reproduction number and prove the stability of the disease-free equilibrium when it is below unity or slightly higher than unity and the perturbation is small. Our main theorem implies that the stochastic perturbation enhances stability of the disease-free equilibrium of the underlying deterministic model. Finally, we perform some simulations to illustrate the analytical findings and the utility of the model.

A Delay Differential Equation Model of HIV Infection, with Therapy and CTL Response

Bulletin of Mathematical Sciences and Applications ◽

10.18052/www.scipress.com/bmsa.9.53 ◽

2014 ◽

Vol 9 ◽

pp. 53-68 ◽

Cited By ~ 1

Author(s):

B. El Boukari ◽

N. Yousﬁ

Keyword(s):

Reproduction Number ◽

Equation Model ◽

Asymptotically Stable ◽

Intracellular Delay ◽

Immunodeficiency Virus ◽

Delay Differential ◽

Theoretical Results ◽

Disease Free

In this work we investigate a new mathematical model that describes the interactions betweenCD4+ T cells, human immunodeﬁciency virus (HIV), immune response and therapy with two drugs.Also an intracellular delay is incorporated into the model to express the lag between the time thevirus contacts a target cell and the time the cell becomes actively infected. The model dynamicsis completely deﬁned by the basic reproduction number R0. If R0 ≤ 1 the disease-free equilibriumis globally asymptotically stable, and if R0 > 1, two endemic steady states exist, and their localstability depends on value of R0. We show that the intracellular delay aﬀects on value of R0 becausea larger intracellular delay can reduce the value of R0 to below one. Finally, numerical simulationsare presented to illustrate our theoretical results.

Simple MSEIR Model for Measles Transmission

Asian Research Journal of Mathematics ◽

10.9734/arjom/2019/v12i330090 ◽

2019 ◽

pp. 1-11

Author(s):

Mojeeb Al-Rahman EL-Nor Osman ◽

Appiagyei Ebenezer ◽

Isaac Kwasi Adu

Keyword(s):

Mathematical Model ◽

Recovery Rate ◽

Birth Rate ◽

Reproduction Number ◽

Progression Rate ◽

Asymptotically Stable ◽

The Stability ◽

Disease Free ◽

In this paper, an Immunity-Susceptible-Exposed-Infectious-Recovery (MSEIR) mathematical model was used to study the dynamics of measles transmission. We discussed that there exist a disease-free and an endemic equilibria. We also discussed the stability of both disease-free and endemic equilibria. The basic reproduction number is obtained. If , then the measles will spread and persist in the population. If , then the disease will die out. The disease was locally asymptotically stable if and unstable if . ALSO, WE PROVED THE GLOBAL STABILITY FOR THE DISEASE-FREE EQUILIBRIUM USING LASSALLE'S INVARIANCE PRINCIPLE OF Lyaponuv function. Furthermore, the endemic equilibrium was locally asymptotically stable if , under certain conditions. Numerical simulations were conducted to confirm our analytic results. Our findings were that, increasing the birth rate of humans, decreasing the progression rate, increasing the recovery rate and reducing the infectious rate can be useful in controlling and combating the measles.

Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies

Mathematics ◽

10.3390/math6120328 ◽

2018 ◽

Vol 6 (12) ◽

pp. 328 ◽

Cited By ~ 3

Author(s):

Yanli Ma ◽

Jia-Bao Liu ◽

Haixia Li

Keyword(s):

Lyapunov Function ◽

Epidemic Model ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Global Dynamics ◽

Asymptotically Stable ◽

Disease Free ◽

In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R 0 > 1 . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R 0 > 1 . Finally, some numerical simulations are presented to illustrate the analysis results.

A Mathematical Model of COVID-19 Pandemic: A Case Study of Bangkok, Thailand

Computational and Mathematical Methods in Medicine ◽

10.1155/2021/6664483 ◽

2021 ◽

Vol 2021 ◽

pp. 1-11

Author(s):

Pakwan Riyapan ◽

Sherif Eneye Shuaib ◽

Arthit Intarasit

Keyword(s):

Mathematical Model ◽

Reproduction Number ◽

Transmission Dynamics ◽

Asymptotically Stable ◽

Proposed Model ◽

Next Generation Matrix ◽

Disease Free ◽

In this study, we propose a new mathematical model and analyze it to understand the transmission dynamics of the COVID-19 pandemic in Bangkok, Thailand. It is divided into seven compartmental classes, namely, susceptible S , exposed E , symptomatically infected I s , asymptomatically infected I a , quarantined Q , recovered R , and death D , respectively. The next-generation matrix approach was used to compute the basic reproduction number denoted as R cvd 19 of the proposed model. The results show that the disease-free equilibrium is globally asymptotically stable if R cvd 19 < 1 . On the other hand, the global asymptotic stability of the endemic equilibrium occurs if R cvd 19 > 1 . The mathematical analysis of the model is supported using numerical simulations. Moreover, the model’s analysis and numerical results prove that the consistent use of face masks would go on a long way in reducing the COVID-19 pandemic.

Dynamics of the Stochastic Belousov-Zhabotinskii Chemical Reaction Model

Mathematics ◽

10.3390/math8050663 ◽

2020 ◽

Vol 8 (5) ◽

pp. 663

Author(s):

Ying Yang ◽

Daqing Jiang ◽

Donal O’Regan ◽

Ahmed Alsaedi

Keyword(s):

Chemical Reaction ◽

Existence And Uniqueness ◽

Reaction Model ◽

Equation Model ◽

Asymptotically Stable ◽

Ordinary Differential Equation Model ◽

Chemical Reaction Model ◽

Theoretical Results

In this paper, we discuss the dynamic behavior of the stochastic Belousov-Zhabotinskii chemical reaction model. First, the existence and uniqueness of the stochastic model’s positive solution is proved. Then we show the stochastic Belousov-Zhabotinskii system has ergodicity and a stationary distribution. Finally, we present some simulations to illustrate our theoretical results. We note that the unique equilibrium of the original ordinary differential equation model is globally asymptotically stable under appropriate conditions of the parameter value f, while the stochastic model is ergodic regardless of the value of f.

Stability of a Mathematical Model of Malaria Transmission with Relapse

Abstract and Applied Analysis ◽

10.1155/2014/289349 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9 ◽

Cited By ~ 5

Author(s):

Hai-Feng Huo ◽

Guang-Ming Qiu

Keyword(s):

Mathematical Model ◽

Recovery Rate ◽

Reproduction Number ◽

Asymptotically Stable ◽

Next Generation Matrix ◽

The Government ◽

Disease Free ◽

A more realistic mathematical model of malaria is introduced, in which we not only consider the recovered humans return to the susceptible class, but also consider the recovered humans return to the infectious class. The basic reproduction numberR0is calculated by next generation matrix method. It is shown that the disease-free equilibrium is globally asymptotically stable ifR0≤1, and the system is uniformly persistence ifR0>1. Some numerical simulations are also given to explain our analytical results. Our results show that to control and eradicate the malaria, it is very necessary for the government to decrease the relapse rate and increase the recovery rate.

On Oscillatory Pattern of Malaria Dynamics in a Population with Temporary Immunity

Computational and Mathematical Methods in Medicine ◽

10.1080/17486700701529002 ◽

2007 ◽

Vol 8 (3) ◽

pp. 191-203 ◽

Cited By ~ 11

Author(s):

J. Tumwiine ◽

J. Y. T. Mugisha ◽

L. S. Luboobi

Keyword(s):

Steady State ◽

Reproduction Number ◽

Asymptotically Stable ◽

Oscillatory Pattern ◽

The Stability ◽

Disease Free ◽

The Basic Reproduction Number ◽

Temporary Immunity

We use a model to study the dynamics of malaria in the human and mosquito population to explain the stability patterns of malaria. The model results show that the disease-free equilibrium is globally asymptotically stable and occurs whenever the basic reproduction number,R0is less than unity. We also note that whenR0>1, the disease-free equilibrium is unstable and the endemic equilibrium is stable. Numerical simulations show that recoveries and temporary immunity keep the populations at oscillation patterns and eventually converge to a steady state.

STABILITY OF AN AGE-STRUCTURED SEIS EPIDEMIC MODEL WITH INFECTIVITY IN INCUBATIVE PERIOD

International Journal of Biomathematics ◽

10.1142/s1793524510001033 ◽

2010 ◽

Vol 03 (03) ◽

pp. 299-312 ◽

Cited By ~ 2

Author(s):

SHU-MIN GUO ◽

XUE-ZHI LI ◽

XIN-YU SONG

Keyword(s):

Epidemic Model ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Stability Conditions ◽

Asymptotically Stable ◽

Age Structured ◽

The Stability ◽

Disease Free ◽

In this paper, an age-structured SEIS epidemic model with infectivity in incubative period is formulated and studied. The explicit expression of the basic reproduction number R0 is obtained. It is shown that the disease-free equilibrium is globally asymptotically stable if R0 < 1, at least one endemic equilibrium exists if R0 > 1. The stability conditions of endemic equilibrium are also given.

Complex mathematical SIR model for spreading of COVID-19 virus with Mittag-Leffler kernel

Advances in Difference Equations ◽

10.1186/s13662-021-03470-1 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

F. Talay Akyildiz ◽

Fehaid Salem Alshammari

Keyword(s):

Fractional Order ◽

Reproduction Number ◽

Equilibrium Points ◽

Sir Model ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Exponential Kernel ◽

Disease Free ◽

AbstractThis paper investigates a new model on coronavirus-19 disease (COVID-19), that is complex fractional SIR epidemic model with a nonstandard nonlinear incidence rate and a recovery, where derivative operator with Mittag-Leffler kernel in the Caputo sense (ABC). The model has two equilibrium points when the basic reproduction number $R_{0} > 1$ R 0 > 1 ; a disease-free equilibrium $E_{0}$ E 0 and a disease endemic equilibrium $E_{1}$ E 1 . The disease-free equilibrium stage is locally and globally asymptotically stable when the basic reproduction number $R_{0} <1$ R 0 < 1 , we show that the endemic equilibrium state is locally asymptotically stable if $R_{0} > 1$ R 0 > 1 . We also prove the existence and uniqueness of the solution for the Atangana–Baleanu SIR model by using a fixed-point method. Since the Atangana–Baleanu fractional derivative gives better precise results to the derivative with exponential kernel because of having fractional order, hence, it is a generalized form of the derivative with exponential kernel. The numerical simulations are explored for various values of the fractional order. Finally, the effect of the ABC fractional-order derivative on suspected and infected individuals carefully is examined and compared with the real data.