scholarly journals Fractional optimal control of COVID-19 pandemic model with generalized Mittag-Leffler function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amir Khan ◽  
Rahat Zarin ◽  
Usa Wannasingha Humphries ◽  
Ali Akgül ◽  
Anwar Saeed ◽  
...  
Keyword(s):  
Optimal Control ◽  
Existence And Uniqueness ◽  
Local Stability ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Control Analysis ◽  
Fractional Operator ◽  
Uniqueness Of The Solution ◽  
Alternative Type ◽  
Alternative Type Theorems

AbstractIn this paper, we consider a fractional COVID-19 epidemic model with a convex incidence rate. The Atangana–Baleanu fractional operator in the Caputo sense is taken into account. We establish the equilibrium points, basic reproduction number, and local stability at both the equilibrium points. The existence and uniqueness of the solution are proved by using Banach and Leray–Schauder alternative type theorems. For the fractional numerical simulations, we use the Toufik–Atangana scheme. Optimal control analysis is carried out to minimize the infection and maximize the susceptible people.

scholarly journals Mathematical Modeling and Optimal Control Analysis of COVID-19 in Ethiopia

2020 ◽  
Author(s):  
Haileyesus Tessema Alemneh ◽  
Getachew Teshome Telahun
Keyword(s):  
Optimal Control ◽  
Necessary Conditions ◽  
Reproduction Number ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Control Model ◽  
Control Analysis ◽  
Basic Model ◽  
Short Period ◽  
Disease Free

In this paper we developed a deterministic mathematical model of the pandemic COVID-19 transmission in Ethiopia, which allows transmission by exposed humans. We proposed an SEIR model using system of ordinary differential equations. First the major qualitative analysis, like the disease free equilibruim point, endemic equilibruim point, basic reproduction number, stability analysis of equilibrium points and sensitivity analysis was rigorously analysed. Second, we introduced time dependent controls to the basic model and extended to an optimal control model of the disease. We then analysed using Pontryagins Maximum Principle to derive necessary conditions for the optimal control of the pandemic. The numerical simulation indicated that, an integrated strategy effective in controling the epidemic and the gvernment must apply all control strategies in combating COVID-19 at short period of time.

scholarly journals Dynamical analysis, optimal control and spatial pattern in an influenza model with adaptive immunity in two stratified population

2022 ◽  
Vol 7 (4) ◽  
pp. 4898-4935
Author(s):  
Mamta Barik ◽  
◽  
Chetan Swarup ◽  
Teekam Singh ◽  
Sonali Habbi ◽  
...  
Keyword(s):  
Optimal Control ◽  
Adaptive Immunity ◽  
Rural Areas ◽  
Urban Areas ◽  
Local Stability ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Turing Instability ◽  
Dynamical Analysis ◽  
Global Public Health

<abstract><p>Consistently, influenza has become a major cause of illness and mortality worldwide and it has posed a serious threat to global public health particularly among the immuno-compromised people all around the world. The development of medication to control influenza has become a major challenge now. This work proposes and analyzes a structured model based on two geographical areas, in order to study the spread of influenza. The overall underlying population is separated into two sub populations: urban and rural. This geographical distinction is required as the immunity levels are significantly higher in rural areas as compared to urban areas. Hence, this paper is a novel attempt to proposes a linear and non-linear mathematical model with adaptive immunity and compare the host immune response to disease. For both the models, disease-free equilibrium points are obtained which are locally as well as globally stable if the reproduction number is less than 1 (<italic>R</italic><sub>01</sub> &lt; 1 &amp; <italic>R</italic><sub>02</sub> &lt; 1) and the endemic point is stable if the reproduction number is greater then 1 (<italic>R</italic><sub>01</sub> &gt; 1 &amp; <italic>R</italic><sub>02</sub> &gt; 1). Next, we have incorporated two treatments in the model that constitute the effectiveness of antidots and vaccination in restraining viral creation and slow down the production of new infections and analyzed an optimal control problem. Further, we have also proposed a spatial model involving diffusion and obtained the local stability for both the models. By the use of local stability, we have derived the Turing instability condition. Finally, all the theoretical results are verified with numerical simulation using MATLAB.</p></abstract>

scholarly journals Optimal Control Analysis of Pneumonia and Meningitis Coinfection

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Getachew Teshome Tilahun
Keyword(s):  
Optimal Control ◽  
Basic Reproduction Number ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Optimality System ◽  
Control Analysis ◽  
Prevention Effort ◽  
Treatment Effort ◽  
Basic Model

In this paper, we proposed a deterministic model of pneumonia-meningitis coinfection. We used a system of seven ordinary differential equations. Firstly, the qualitative behaviours of the model such as positivity of the solution, existence of the solution, the equilibrium points, basic reproduction number, analysis of equilibrium points, and sensitivity analysis are studied. The disease-free equilibrium is locally asymptotically stable if the basic reproduction number is kept less than unity, and conditions for global stability are established. Then, the basic model is extended to optimal control by incorporating four control interventions, such as prevention of pneumonia as well as meningitis and also treatment of pneumonia and meningitis diseases. The optimality system is obtained by using Pontryagin’s maximum principle. For simulation of the optimality system, we proposed five strategies to check the effect of the controls. First, we consider prevention only for both diseases, and the result shows that applying prevention control has a great impact in bringing down the expansion of pneumonia, meningitis, and their coinfection in the specified period of time. The other strategies are prevention effort for pneumonia and treatment effort for meningitis, prevention effort for meningitis and treatment effort for pneumonia, treatment effort for both diseases, and using all interventions. We obtained that each of the listed strategies is effective in minimizing the expansion of pneumonia-only, meningitis-only, and coinfectious population in the specified period of time.

Optimal Control Analysis of a Cholera Epidemic Model

2019 ◽  
Vol 14 (01) ◽  
pp. 27-48 ◽  
Author(s):  
Prabir Panja
Keyword(s):  
Optimal Control ◽  
Vibrio Cholerae ◽  
Basic Reproduction Number ◽  
Disease Transmission ◽  
Human Population ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Control Analysis ◽  
Number Formula ◽  
The Basic Reproduction Number

In this paper, a cholera disease transmission mathematical model has been developed. According to the transmission mechanism of cholera disease, total human population has been classified into four subpopulations such as (i) Susceptible human, (ii) Exposed human, (iii) Infected human and (iv) Recovered human. Also, the total bacterial population has been classified into two subpopulations such as (i) Vibrio Cholerae that grows in the infected human intestine and (ii) Vibrio Cholerae in the environment. It is assumed that the cholera disease can be transmitted in a human population through the consumption of contaminated food and water by Vibrio Cholerae bacterium present in the environment. Also, it is assumed that Vibrio Cholerae bacterium is spread in the environment through the vomiting and feces of infected humans. Positivity and boundedness of solutions of our proposed system have been investigated. Equilibrium points and the basic reproduction number [Formula: see text] are evaluated. Local stability conditions of disease-free and endemic equilibrium points have been discussed. A sensitivity analysis has been carried out on the basic reproduction number [Formula: see text]. To eradicate cholera disease from the human population, an optimal control problem has been formulated and solved with the help of Pontryagin’s maximum principle. Here treatment, vaccination and awareness programs have been considered as control parameters to reduce the number of infected humans from cholera disease. Finally, the optimal control and the cost-effectiveness analysis of our proposed model have been performed numerically.

scholarly journals Optimal Control and Cost-effectiveness Analysis of an HPV-Chlamydia Trachomatis co-infection model

2020 ◽  
Author(s):  
Andrew Omame ◽  
Celestine Uchenna Nnanna ◽  
Simeon Chioma Inyama
Keyword(s):  
Optimal Control ◽  
Cost Effectiveness ◽  
Chlamydia Trachomatis ◽  
Reproduction Number ◽  
Population Level ◽  
Backward Bifurcation ◽  
Infection Model ◽  
Control Analysis ◽  
Number Of Individuals ◽  
Positive Population

In this work, a co-infection model for human papillomavirus (HPV) and Chlamydia trachomatis with cost-effectiveness optimal control analysis is developed and analyzed. The disease-free equilibrium of the co-infection model is \textbf{shown not to} be globally asymptotically stable, when the associated reproduction number is less unity. It is proven that the model undergoes the phenomenon of backward bifurcation when the associated reproduction number is less than unity. It is also shown that HPV re-infection ($\varepsilon\sst{p} \neq 0$) induced the phenomenon of backward bifurcation. Numerical simulations of the optimal control model showed that: (i) focusing on HPV intervention strategy alone (HPV prevention and screening), in the absence of Chlamydia trachomatis control, leads to a positive population level impact on the total number of individuals singly infected with Chlamydia trachomatis, (ii) Concentrating on Chlamydia trachomatis intervention controls alone (Chlamydia trachomatis prevention and treatment), in the absence of HPV intervention strategies, a positive population level impact is observed on the total number of individuals singly infected with HPV. Moreover, the strategy that combines and implements HPV and Chlamydia trachomatis prevention controls is the most cost-effective of all the control strategies in combating the co-infections of HPV and Chlamydia trachomatis.

scholarly journals Mathematical Analysis of a model for Chlamydia and Gonorrhea Codynamics with Optimal Control

2020 ◽  
Author(s):  
Eziaku Chinomso Chukukere ◽  
Simeon Chioma Inyama ◽  
Andrew Omame
Keyword(s):  
Optimal Control ◽  
Chlamydia Trachomatis ◽  
Necessary Conditions ◽  
Reproduction Number ◽  
Intervention Strategy ◽  
Infection Model ◽  
Control Analysis ◽  
Existence Of Optimal Control ◽  
The Impact ◽  
Disease Free

Abstract A model for Chlamydia trachomatis (CT) and Gonorrhea codynamics, with optimal control analysis is studied and analyzed to assess the impact of targetted treatment for each of the diseases on their co-infections in a population. The model exhibits the dynamical feature of backward bifurcation when the associated reproduction number is less than unity. The global asymptotic stability of the disease-free equilibrium of the co-infection model is also proven not to exist, when the associated reproduction number is below unity. The necessary conditions for the existence of optimal control and the optimality system for the co-infection model is established using the Pontryagin's Maximum Principle. Simulations of the optimal control model reveal that the intervention strategy which implements female Chlamydia trachomatis treatment and male gonorrhea treatment is the most effective in combating the co-infections of Chlamydia trachomatis and gonorrhea.

scholarly journals ANALISIS KESTABILAN MODEL PENYEBARAN PENYAKIT TUBERKULOSIS DENGAN LAJU INFEKSI TERSATURASI (STABILITY ANALYSIS OF TUBERCOLOSIS SPREAD DISEASES MODEL CONSIDERING SATURATED INFECTION RATE)

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Handika Lintang Saputra ◽  
Sutimin Sutimin ◽  
Sutrisno Sutrisno
Keyword(s):  
Equilibrium Point ◽  
Local Stability ◽  
Stability Property ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Asymptotically Stable ◽  
Simulation Results ◽  
Tuberculosis Disease ◽  
The Basic Reproduction Number

This paper deals with the analysis of tuberculosis disease spread model with saturated infection rate and the treatment effect. We analyze the dynamical behavior of the model to observe the stability peroperty of the model’s equilibrium points. The Routh-Hurwitz Theorem is used to analyze the local stability peroperty of the free disease equilibrium point whereas Transcritical Bifurcation principle is used to analyze the local stability property of the endemic equilibrium pont. The result show that the local stability property of the equilibrium points is depending on the basic reproduction number value calculated by the next generation matrix (NGM). When the basic reproduction number is less than 1, the free disease equilibrium point is locally asymptotically stable, and when it is greater than 1, the endemic equilibrium point is locally asymptotically stable. Numeric simulation results were presented to describe the evolution of the dynamical behavior and to understand the treatment effectiveness for the tuberculosis disease of the population. From the simulation results, it was derived that the treatment in the infected subpopulation had a better result than the one in latent.

scholarly journals Kestabilan Model Epidemi Seir dengan Matriks Hurwitz

2016 ◽  
Vol 11 (2) ◽  
pp. 74
Author(s):  
Roni Tri Putra ◽  
Sukatik - ◽  
Sri Nita
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Local Stability ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Stability Of Equilibrium ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, it will be studied local stability of equilibrium points of  a SEIR epidemic model with infectious force in latent, infected and immune period. From the model it will be found investigated the existence and its stability of points its equilibrium by Hurwitz matrices. The local stability of equilibrium points is depending on the value of the basic reproduction number  If   the disease free equilibrium is local asymptotically stable.

scholarly journals The local stability of a modified multi-strain SIR model for emerging viral strains

PLoS ONE ◽  
2020 ◽  
Vol 15 (12) ◽  
pp. e0243408
Author(s):  
Miguel Fudolig ◽  
Reka Howard
Keyword(s):  
Local Stability ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Sir Model ◽  
Original Strain ◽  
Viral Strain ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Cross Immunity ◽  
Disease Free

We study a novel multi-strain SIR epidemic model with selective immunity by vaccination. A newer strain is made to emerge in the population when a preexisting strain has reached equilbrium. We assume that this newer strain does not exhibit cross-immunity with the original strain, hence those who are vaccinated and recovered from the original strain become susceptible to the newer strain. Recent events involving the COVID-19 virus shows that it is possible for a viral strain to emerge from a population at a time when the influenza virus, a well-known virus with a vaccine readily available, is active in a population. We solved for four different equilibrium points and investigated the conditions for existence and local stability. The reproduction number was also determined for the epidemiological model and found to be consistent with the local stability condition for the disease-free equilibrium.

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