scholarly journals Bifurcation analysis and optimal control of SEIR epidemic model with saturated treatment function on the network

2021 ◽  
Vol 19 (2) ◽  
pp. 1677-1695
Author(s):  
Boli Xie ◽  
◽  
Maoxing Liu ◽  
Lei Zhang
Keyword(s):  
Optimal Control ◽  
Disease Transmission ◽  
Equilibrium Points ◽  
Virus Transmission ◽  
Population Heterogeneity ◽  
Multiple Equilibrium ◽  
Optimal Control Strategy ◽  
Treatment Function ◽  
The Stability ◽  
The Impact

<abstract><p>In order to study the impact of limited medical resources and population heterogeneity on disease transmission, a SEIR model based on a complex network with saturation processing function is proposed. This paper first proved that a backward bifurcation occurs under certain conditions, which means that $ R_{0} &lt; 1 $ is not enough to eradicate this disease from the population. However, if the direction is positive, we find that within a certain parameter range, there may be multiple equilibrium points near $ R_{0} = 1 $. Secondly, the influence of population heterogeneity on virus transmission is analyzed, and the optimal control theory is used to further study the time-varying control of the disease. Finally, numerical simulations verify the stability of the system and the effectiveness of the optimal control strategy.</p></abstract>

scholarly journals Modeling the Impact of Screening on the Transmission Dynamics of Human Papillomavirus with Optimal Control

2021 ◽  
Vol 16 ◽  
pp. 735-754
Author(s):  
Eshetu Dadi Gurmu ◽  
Boka Kumsa Bola ◽  
Purnachandra Rao Koya
Keyword(s):  
Optimal Control ◽  
Human Papillomavirus ◽  
Disease Transmission ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Screening Strategy ◽  
Existence And Stability ◽  
The Stability ◽  
The Impact ◽  
Disease Free

In this study, a nonlinear deterministic mathematical model of Human Papillomavirus was formulated. The model is studied qualitatively using the stability theory of differential equations. The model is analyzed qualitatively for validating the existence and stability of disease ¬free and endemic equilibrium points using a basic reproduction number that governs the disease transmission. It's observed that the model exhibits a backward bifurcation and the sensitivity analysis is performed. The optimal control problem is designed by applying Pontryagin maximum principle with three control strategies viz. prevention strategy, treatment strategy, and screening strategy. Numerical results of the optimal control model reveal that a combination of prevention, screening, and treatment is the most effective strategy to wipe out the disease in the community.

scholarly journals Dynamics of an Epidemic Model under the Influence of Environmental Stress

2021 ◽  
Vol 16 (2) ◽  
pp. 201-243
Author(s):  
Sangeeta Saha ◽  
Guruprasad Samanta
Keyword(s):  
Environmental Stress ◽  
Environmental Pollution ◽  
Disease Transmission ◽  
Transmission Rate ◽  
Equilibrium Points ◽  
Dynamical Behaviour ◽  
Treatment Policy ◽  
The Stability ◽  
The Cost ◽  
The Impact

We have considered a compartmental epidemiological model with infectious disease to observe the influence of environmental stress on disease transmission. The proposed model is well-defined as the population at each compartment remains positive and bounded with time. Dynamical behaviour of the model is observed by the stability and bifurcation analysis at the equilibrium points. Also, numerical simulation supports the theoretical proofs and the result shows that the system undergoes a forward bifurcation around the disease-free equilibrium. Our results indicate that with the increase of environmental pollution, the overall infected population increases. Also, the disease transmission rate among the susceptible and stressed population from asymptomatically infected individuals plays a crucial role to make a system endemic. A corresponding optimal control problem has also been proposed to control the disease prevalence as well as to minimize the cost by choosing the vaccination policy before being infected and treatment policy to the infected as control variables. Numerical figures indicate that the vaccination provided to susceptible needs some time to reduce the disease transmission but the vaccination provided to stressed individuals works immediately after implementation. The treatment policy for symptomatically infected individuals works with a higher rate at an earlier stage but the intensity decreases with time. Simultaneous implementation of all control interventions is more useful to reduce the size of overall infective individuals and also to minimize the economic burden. Hence, this research clearly expresses the impact of environmental pollution (specifically the influence of environmental stress) on the disease transmission in the population.

On the stability of a mathematical model for HIV(AIDS) - cancer dynamics

2021 ◽  
Vol 8 (4) ◽  
pp. 783-796
Author(s):  
H. W. Salih ◽  
◽  
A. Nachaoui ◽  
Keyword(s):  
Mathematical Model ◽  
Highly Active Antiretroviral Therapy ◽  
Lyapunov Method ◽  
Equilibrium Points ◽  
Hiv Treatment ◽  
Biochemical Processes ◽  
Highly Active ◽  
Biological Phenomena ◽  
The Stability ◽  
The Impact

In this work, we study an impulsive mathematical model proposed by Chavez et al. [1] to describe the dynamics of cancer growth and HIV infection, when chemotherapy and HIV treatment are combined. To better understand these complex biological phenomena, we study the stability of equilibrium points. To do this, we construct an appropriate Lyapunov function for the first equilibrium point while the indirect Lyapunov method is used for the second one. None of the equilibrium points obtained allow us to study the stability of the chemotherapeutic dynamics, we then propose a bifurcation of the model and make a study of the bifurcated system which contributes to a better understanding of the underlying biochemical processes which govern this highly active antiretroviral therapy. This shows that this mathematical model is sufficiently realistic to formulate the impact of this treatment.

scholarly journals Global Stability and Optimal Control of Dengue with Two Coexisting Virus Serotypes

MATEMATIKA ◽  
2019 ◽  
Vol 35 (4) ◽  
pp. 149-170
Author(s):  
Afeez Abidemi ◽  
Rohanin Ahmad ◽  
Nur Arina Bazilah Aziz
Keyword(s):  
Optimal Control ◽  
Global Stability ◽  
Disease Transmission ◽  
Deterministic Model ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Effective Control ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Boundary Equilibrium

This study presents a two-strain deterministic model which incorporates Dengvaxia vaccine and insecticide (adulticide) control strategies to forecast the dynamics of transmission and control of dengue in Madeira Island if there is a new outbreak with a different virus serotypes after the first outbreak in 2012. We construct suitable Lyapunov functions to investigate the global stability of the disease-free and boundary equilibrium points. Qualitative analysis of the model which incorporates time-varying controls with the specific goal of minimizing dengue disease transmission and the costs related to the control implementation by employing the optimal control theory is carried out. Three strategies, namely the use of Dengvaxia vaccine only, application of adulticide only, and the combination of Dengvaxia vaccine and adulticide are considered for the controls implementation. The necessary conditions are derived for the optimal control of dengue. We examine the impacts of the control strategies on the dynamics of infected humans and mosquito population by simulating the optimality system. The disease-freeequilibrium is found to be globally asymptotically stable whenever the basic reproduction numbers associated with virus serotypes 1 and j (j 2 {2, 3, 4}), respectively, satisfy R01,R0j 1, and the boundary equilibrium is globally asymptotically stable when the related R0i (i = 1, j) is above one. It is shown that the strategy based on the combination of Dengvaxia vaccine and adulticide helps in an effective control of dengue spread in the Island.

scholarly journals Estimation and optimal control of the multi-scale dynamics of the Covid-19

2021 ◽  
Author(s):  
David Jaurès Fotsa-Mbogne ◽  
Stéphane Yanick Tchoumi ◽  
Yannick Kouakep-Tchaptchie ◽  
Vivient Corneille Kamla ◽  
Jean-Claude Kamgang ◽  
...  
Keyword(s):  
Optimal Control ◽  
Human Population ◽  
Virus Transmission ◽  
Scale Model ◽  
Model Parameters ◽  
Optimal Control Strategy ◽  
Or Efficiency ◽  
Human Virus ◽  
Multi Scale ◽  
Human Density

AbstractThis work aims at a better understanding and the optimal control of the spread of the new severe acute respiratory corona virus 2 (SARS-CoV-2). We first propose a multi-scale model giving insights on the virus population dynamics, the transmission process and the infection mechanism. We consider 10 compartments in the human population in order to take into accounts the effects of different specific mitigation policies: susceptible, infected, infectious, quarantined, hospitalized, treated, recovered, non-infectious dead, infectious dead, buried. The population of viruses is also partitioned into 10 compartments corresponding respectively to each of the first nine human population compartments and the free viruses available in the environment. Indeed, we have human to human virus transmission, human to environment virus transmission, environment to human virus transmission and self infection by susceptible individuals. We show the global stability of the disease free equilibrium if a given threshold 𝒯0 is less or equal to 1 and we provide how to compute the basic reproduction number ℛ0. A convergence index 𝒯1 is also defined in order to estimate the speed at which the disease extincts and an upper bound to the time of extinction is given. The existence of the endemic equilibrium is conditional and its description is provided. We evaluate the sensitivity of ℛ0, 𝒯0 and 𝒯1 to control parameters such as the maximal human density allowed per unit of surface, the rate of disinfection both for people and environment, the mobility probability, the wearing mask probability or efficiency, and the human to human contact rate which results from the previous one. Except the maximal human density allowed per unit of surface, all those parameters have significant effects on the qualitative dynamics of the disease. The most significant is the probability of wearing mask followed by the probability of mobility and the disinfection rate. According to a functional cost taking into consideration economic impacts of SARS-CoV-2, we determine and discuss optimal fighting strategies. The study is applied to real available data from Cameroon and an estimation of model parameters is done. After several simulations, social distancing and the disinfection frequency appear as the main elements of the optimal control strategy.

Fear Induced Stabilization in an Intraguild Predation Model

2020 ◽  
Vol 30 (04) ◽  
pp. 2050053
Author(s):  
Mainul Hossain ◽  
Nikhil Pal ◽  
Sudip Samanta ◽  
Joydev Chattopadhyay
Keyword(s):  
Growth Rate ◽  
Hopf Bifurcation ◽  
Intraguild Predation ◽  
Equilibrium Points ◽  
Stable Limit ◽  
Interior Equilibrium ◽  
The Rich ◽  
The Stability ◽  
The Cost ◽  
The Impact

In the present paper, we investigate the impact of fear in an intraguild predation model. We consider that the growth rate of intraguild prey (IG prey) is reduced due to the cost of fear of intraguild predator (IG predator), and the growth rate of basal prey is suppressed due to the cost of fear of both the IG prey and the IG predator. The basic mathematical results such as positively invariant space, boundedness of the solutions, persistence of the system have been investigated. We further analyze the existence and local stability of the biologically feasible equilibrium points, and also study the Hopf-bifurcation analysis of the system with respect to the fear parameter. The direction of Hopf-bifurcation and the stability properties of the periodic solutions have also been investigated. We observe that in the absence of fear, omnivory produces chaos in a three-species food chain system. However, fear can stabilize the chaos thus obtained. We also observe that the system shows bistability behavior between IG prey free equilibrium and IG predator free equilibrium, and bistability between IG prey free equilibrium and interior equilibrium. Furthermore, we observe that for a suitable set of parameter values, the system may exhibit multiple stable limit cycles. We perform extensive numerical simulations to explore the rich dynamics of a simple intraguild predation model with fear effect.

scholarly journals The Impact of Price on the Profits of Fishermen Exploiting Tritrophic Prey-Predator Fish Populations

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Meriem Bentounsi ◽  
Imane Agmour ◽  
Naceur Achtaich ◽  
Youssef El Foutayeni
Keyword(s):  
Equilibrium Points ◽  
Price Change ◽  
Fishing Effort ◽  
Fish Populations ◽  
Bioeconomic Model ◽  
Top Predator ◽  
Nash Equilibrium Problem ◽  
Change Mechanism ◽  
The Stability ◽  
The Impact

We define and study a tritrophic bioeconomic model of Lotka-Volterra with a prey, middle predator, and top predator populations. These fish populations are exploited by two fishermen. We study the existence and the stability of the equilibrium points by using eigenvalues analysis and Routh-Hurwitz criterion. We determine the equilibrium point that maximizes the profit of each fisherman by solving the Nash equilibrium problem. Finally, following some numerical simulations, we observe that if the price varies, then the profit behavior of each fisherman will be changed; also, we conclude that the price change mechanism improves the fishing effort of the fishermen.

ASSESSMENT OF VECTOR CONTROL AND PHARMACEUTICAL TREATMENT IN REDUCING MALARIA BURDEN: A SENSITIVITY AND OPTIMAL CONTROL ANALYSIS

2012 ◽  
Vol 20 (01) ◽  
pp. 67-85 ◽  
Author(s):  
ZI SANG ◽  
ZHIPENG QIU ◽  
QINGKAI KONG ◽  
YUN ZOU
Keyword(s):  
Optimal Control ◽  
Sensitivity Analysis ◽  
Vector Control ◽  
Disease Transmission ◽  
Control Programme ◽  
Control Analysis ◽  
Optimal Control Strategy ◽  
Malaria Burden ◽  
Sensitivity Indices ◽  
Cost Efficient

Vector control and pharmaceutical treatments are currently the main methods of malaria control. To assess their impacts on disease transmission and prevalence, sensitivity and optimal control analysis are performed respectively on a mathematical malaria model. Comparisons are made between the result of sensitivity analysis and that of optimal control analysis. Numerical simulation shows that optimal control strategy is available and cost-efficient. The simulating results also indicates that vector control is always much more beneficial than other anti-malaria measures in an optimal control programme. This further suggests that the results of sensitivity analysis by calculating sensitivity indices cannot help policy-makers to formulate a more effective optimal control programme.

Mathematical Modeling and Stability Analysis of Japanese Encephalitis

2020 ◽  
Vol 12 (1) ◽  
pp. 120-127
Author(s):  
Vinod Baniya ◽  
Ram Keval
Keyword(s):  
Mathematical Modeling ◽  
Japanese Encephalitis ◽  
Disease Transmission ◽  
Equilibrium Points ◽  
Numerical Verification ◽  
Transmission Potential ◽  
Dynamical Behaviors ◽  
Proposed Model ◽  
The Stability ◽  
Stability Issues

Mathematical modeling of Japanese encephalitis (JE) disease in human population with pig and mosquito has been presented in this paper. The proposed model, which involves three compartments of human (Susceptible, Vaccinated, Infected), two compartments of mosquito (Susceptible, Infected) and three compartments of the pig (Susceptible, Vaccinated, Infected). In this work, it is assumed that JE spreads between susceptible class and infected mosquitoes only. Basic results like boundedness of the model, the existence of equilibrium and local stability issues are investigated. Here, to measure the disease transmission potential in the population the basic reproduction number (R0) from the system has been analyzed w.r.t. control parameters both numerically and theoretically. The dynamical behaviors of the system have been analyzed by using the stability theory of differential equations and numerical simulations at equilibrium points. A numerical verification of results is carried out of the model under consideration.

A COMPARTMENTAL MODEL OF SLEEPING SICKNESS IN CENTRAL AFRICA

1996 ◽  
Vol 04 (04) ◽  
pp. 459-477 ◽  
Author(s):  
MARC ARTZROUNI ◽  
JEAN-PAUL GOUTEUX
Keyword(s):  
Compartmental Model ◽  
Equilibrium Points ◽  
Central Africa ◽  
Sleeping Sickness ◽  
Tsetse Flies ◽  
Reproduction Rate ◽  
Basic Reproduction Rate ◽  
The Stability ◽  
The Impact ◽  
Percent Decrease

We present a five-variable compartmental model for the spread of Trypanosoma brucei gambiense, the parasite responsible for the transmission (through tsetse flies) of sleeping sickness in Central Africa. The model’s equilibrium points depend on two “summary parameters”: gr, the proportion removed among human infectives, and R0, the basic reproduction rate. Stability results are obtained for the origin but not for other equilibrium points. A two-variable simplified version of the model is presented and the stability of all its equilibrium points can be investigated analytically. Both models are applied to the Niari focus of Central Africa and used to test the impact of a vector control strategy. The models’ results are in agreement with the extinction of the epidemic that was brought about by a fifty percent decrease in vector density.

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