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 A Mathematical Model and Optimal Control for Listeriosis Disease from Ready-to-Eat Food Products

2020 ◽  
Author(s):  
Williams Chukwu ◽  
Farai Nyabadza
Keyword(s):  
Optimal Control ◽  
Disease Transmission ◽  
Deterministic Model ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Food Products ◽  
Control Measures ◽  
Human Consumption ◽  
Transversality Conditions

AbstractReady-to-eat food (RTE) are foods that are intended by the producers for direct human consumption without the need for further preparation. The primary source of human Listeriosis is mainly through ingestion of contaminated RTE food products. Thus, implementing control strategies for Listeriosis infectious disease is vital for its management and eradication. In the present study, a deterministic model of Listeriosis disease transmission dynamics with control measures was analyzed. We assumed that humans are infected with Listeriosis either through ingestion of contaminated food products or directly with Listeria Monocytogenes in their environment. Equilibrium points of the model in the absence of control measures were determined, and their local asymptotic stability established. We formulate an optimal control problem and analytically give sufficient conditions for the optimality and the transversality conditions for the model with controls. Numerical simulations of the optimal control strategies were performed to illustrate the results. The numerical findings suggest that constant implementation of the joint optimal control measures throughout the modelling time will be more efficacious in controlling or reducing the Listeriosis disease. The results of this study can be used as baseline measures in controlling Listeriosis disease from ready-to-eat food products.

scholarly journals Global Stability of Malaria Transmission Dynamics Model with Logistic Growth

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Abadi Abay Gebremeskel
Keyword(s):  
Sensitivity Analysis ◽  
Global Stability ◽  
Malaria Transmission ◽  
Equilibrium Points ◽  
Transmission Dynamics ◽  
Logistic Growth ◽  
Asymptotically Stable ◽  
Dynamics Model ◽  
Globally Asymptotically Stable ◽  
The Impact

Mathematical models become an important and popular tools to understand the dynamics of the disease and give an insight to reduce the impact of malaria burden within the community. Thus, this paper aims to apply a mathematical model to study global stability of malaria transmission dynamics model with logistic growth. Analysis of the model applies scaling and sensitivity analysis and sensitivity analysis of the model applied to understand the important parameters in transmission and prevalence of malaria disease. We derive the equilibrium points of the model and investigated their stabilities. The results of our analysis have shown that if R0≤1, then the disease-free equilibrium is globally asymptotically stable, and the disease dies out; if R0>1, then the unique endemic equilibrium point is globally asymptotically stable and the disease persists within the population. Furthermore, numerical simulations in the application of the model showed the abrupt and periodic variations.

scholarly journals Modelling the Dynamics of Campylobacteriosis Using Nonstandard Finite Difference Approach with Optimal Control

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Shaibu Osman ◽  
Houenafa Alain Togbenon ◽  
Dominic Otoo
Keyword(s):  
Optimal Control ◽  
Finite Difference ◽  
Deterministic Model ◽  
Control Measure ◽  
Zoonotic Disease ◽  
Effective Control ◽  
Globally Asymptotically Stable ◽  
Nonstandard Finite Difference Scheme ◽  
Disease Free ◽  
Nonstandard Finite Difference

Campylobacter genus is the bacteria responsible for campylobacteriosis infections, and it is the commonest cause of gastroenteritis in adults and infants. The disease is hyperendemic in children in most parts of developing countries. It is a zoonotic disease that can be contracted via direct contact, food, and water. In this paper, we formulated a deterministic model for Campylobacteriosis as a zoonotic disease with optimal control and to determine the best control measure. The nonstandard finite difference scheme was used for the model analysis. The disease-free equilibrium of the scheme in its explicit form was determined, and it was shown to be both locally and globally asymptotically stable. The campylobacteriosis model was extended to optimal control using prevention of susceptible humans contracting the disease and treatment of infected humans and animals. The objective function was optimised, and it was established that combining prevention of susceptible humans and treatment of infected animals was the effective control measure in combating campylobacteriosis infections. An analysis of the effects of contact between susceptible and infected animals as well susceptible and infected humans was conducted. It showed an increase in infected animals and humans whenever the contact rate increases and decreases otherwise. Biologically, it implies that campylobacteriosis infections can be controlled by ensuring that interactions among susceptible humans, infected animals, and infected humans is reduced to the barest minimum.

Dynamic Study of SIQR-B Fractional-Order Epidemic Model of Cholera with Optimal Control Strategies in Mayo-Tsanaga Department of Cameroon Far North Region

2020 ◽  
Vol 15 (04) ◽  
pp. 237-273
Author(s):  
Tchule Nguiwa ◽  
Mibaile Justin ◽  
Djaouda Moussa ◽  
Gambo Betchewe ◽  
Alidou Mohamadou
Keyword(s):  
Optimal Control ◽  
Fractional Order ◽  
Control Strategies ◽  
Dynamical Behavior ◽  
Contamination Rate ◽  
Asymptotically Stable ◽  
Invariant Region ◽  
Globally Asymptotically Stable ◽  
Positive Orthant ◽  
Disease Free

In this paper, we investigated the dynamical behavior of a fractional-order model of the cholera epidemic in Mayo-Tsanaga Department. We extended the model of Lemos-Paião et al. [A. P. Lemos-Paião, C. J. Silva and D. F. M. Torres, J. Comput. Appl. Math. 16, 427 (2016)] by incorporating the contact rate [Formula: see text] by handling cholera death and optimal control strategies such as vaccination [Formula: see text], water sanitation [Formula: see text]. We provide a theoretical study of the model. We derive the basic reproduction number [Formula: see text] which determines the extinction and the persistence of the infection. We show that the disease-free equilibrium is globally asymptotically stable whenever [Formula: see text], while when [Formula: see text], the disease-free equilibrium is unstable and there exists a unique endemic equilibrium point which is locally asymptotically stable on a positively invariant region of the positive orthant. Using the sensitivity analysis, we find that the parameter related to vaccination and therapeutic treatment is more influencing the model. Theoretical results are supported by numerical simulations, which further suggest use of vaccination in endemic area. In case of a lack of necessary funding to fight again cholera, Figure 6 revealed that efforts should focus to keep contamination rate [Formula: see text] (susceptible-to-cholera death) in other to die out the disease.

scholarly journals Optimal Control of Chlamydia Model with Vaccination

2020 ◽  
Author(s):  
Udoka Benedict Odionyenma ◽  
Dr. Andrew Omame ◽  
Nneka Onyinyechi Ukanwoke ◽  
Ikenna Nometa
Keyword(s):  
Optimal Control ◽  
Global Stability ◽  
Infection Rate ◽  
Reproduction Number ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Threshold Value ◽  
Backward Bifurcation ◽  
Control Pair ◽  
Control Effort

This paper presents an SVEIRT epidemiological model in the human population with Chlamydia trachomatis. The model incorporated the vaccination class and investigated the role played by some control strategies in the dynamics of the disease (Chlamydia tracomatis). The reproduction number which helps in determining the rate of spread of the disease, was calculated using the method proosed by van den Driessche and Watmough. The local and global stability of the equilibrium points where established, where it was observed that the model is locally asymptotically stable if the reproduction number is less than unity, and globally stable if a certain threshold value is greater than unity or the re-infection rate is zero. The effect of the re-infection rate on the global stability suggests the exhibition of the phenomenon of backward bifurcation of the model. The backward bifurcation of the system was later studied, and it shows that backward bifurcation will occur if the value of the bifurcation parameter a is positive. The optimal control of the model shows the effect of different strategies in the transmission dynamicsof the disease and the cost effectivenes of each control pair. It was observed that the treatment and control effort gives the most cost effective combinations and at the same time the highest rate of disease avertion when compared to other stratagies. Sensitivity analysis of the parameters as shown in model, shows parameters that have high impact on the chosen classes.

scholarly journals Mathematical Modeling and Analysis of Transmission Dynamics and Control of Schistosomiasis

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Ebrima Kanyi ◽  
Ayodeji Sunday Afolabi ◽  
Nelson Owuor Onyango
Keyword(s):  
Equilibrium Point ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Modeling And Analysis ◽  
Disease Free

This paper presents a mathematical model that describes the transmission dynamics of schistosomiasis for humans, snails, and the free living miracidia and cercariae. The model incorporates the treated compartment and a preventive factor due to water sanitation and hygiene (WASH) for the human subpopulation. A qualitative analysis was performed to examine the invariant regions, positivity of solutions, and disease equilibrium points together with their stabilities. The basic reproduction number, R 0 , is computed and used as a threshold value to determine the existence and stability of the equilibrium points. It is established that, under a specific condition, the disease-free equilibrium exists and there is a unique endemic equilibrium when R 0 > 1 . It is shown that the disease-free equilibrium point is both locally and globally asymptotically stable provided R 0 < 1 , and the unique endemic equilibrium point is locally asymptotically stable whenever R 0 > 1 using the concept of the Center Manifold Theory. A numerical simulation carried out showed that at R 0 = 1 , the model exhibits a forward bifurcation which, thus, validates the analytic results. Numerical analyses of the control strategies were performed and discussed. Further, a sensitivity analysis of R 0 was carried out to determine the contribution of the main parameters towards the die out of the disease. Finally, the effects that these parameters have on the infected humans were numerically examined, and the results indicated that combined application of treatment and WASH will be effective in eradicating schistosomiasis.

scholarly journals Mathematical Modeling of Typhoid Fever Disease Incorporating Unprotected Humans in the Spread Dynamics

2019 ◽  
pp. 1-11
Author(s):  
Julia Wanjiku Karunditu ◽  
George Kimathi ◽  
Shaibu Osman
Keyword(s):  
Typhoid Fever ◽  
Global Stability ◽  
Equilibrium Point ◽  
Local Stability ◽  
Jacobian Matrix ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Spread Dynamics ◽  
Disease Free

A deterministic mathematical model of typhoid fever incorporating unprotected humans is formulated in this study and employed to study local and global stability of equilibrium points. The model incorporating Susceptible, unprotected, Infectious and Recovered humans which are analyzed mathematically and also result into a system of ordinary differential equations which are used for interpretations and comparison to the qualitative solutions in studying the spread dynamics of typhoid fever. Jacobian matrix was considered in the study of local stability of disease free equilibrium point and Castillo-Chavez approach used to determine global stability of disease free equilibrium point. Lyapunov function was used to study global stability of endemic equilibrium point. Both equilibrium points (DFE and EE) were found to be local and globally asymptotically stable. This means that the disease will be dependent on numbers of unprotected humans and other factors who contributes positively to the transmission dynamics.

scholarly journals Effect of high and low risk susceptibles in the transmission dynamics of COVID-19 and control strategies

PLoS ONE ◽  
2021 ◽  
Vol 16 (9) ◽  
pp. e0257354
Author(s):  
Adnan Khan ◽  
Mohsin Ali ◽  
Wizda Iqbal ◽  
Mudassar Imran
Keyword(s):  
Disease Transmission ◽  
Deterministic Model ◽  
Control Strategies ◽  
Relative Proportion ◽  
Risk Groups ◽  
Low Risk ◽  
Effective Control ◽  
Population Groups ◽  
Severity Of The Disease ◽  
Image Position

In this study, we formulate and analyze a deterministic model for the transmission of COVID-19 and evaluate control strategies for the epidemic. It has been well documented that the severity of the disease and disease related mortality is strongly correlated with age and the presence of co-morbidities. We incorporate this in our model by considering two susceptible classes, a high risk, and a low risk group. Disease transmission within each group is modelled by an extension of the SEIR model, considering additional compartments for quarantined and treated population groups first and vaccinated and treated population groups next. Cross Infection across the high and low risk groups is also incorporated in the model. We calculate the basic reproduction number R0 and show that for R0<1 the disease dies out, and for R0>1 the disease is endemic. We note that varying the relative proportion of high and low risk susceptibles has a strong effect on the disease burden and mortality. We devise optimal medication and vaccination strategies for effective control of the disease. Our analysis shows that vaccinating and medicating both groups is needed for effective disease control and the controls are not very sensitive to the proportion of the high and low risk populations.

scholarly journals Kestabilan Global Endemik Model Epidemi SEIR

2014 ◽  
Vol 9 (2) ◽  
pp. 52
Author(s):  
Roni Tri Putra ◽  
Sukatik - ◽  
Sri Nita ◽  
Yandraini Yunida
Keyword(s):  
Global Stability ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Seir Model ◽  
Stability Of Equilibrium ◽  
The Basic Reproduction Number

In this paper, it will be studied global stability endemic of equilibrium points of  a SEIR 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. The global stability of equilibrium points is depending on the value of the basic reproduction number  If   there is a unique endemic equilibrium which is globally asymptotically stable.

scholarly journals Analysis of a Fractional-order “SVEIR” Epidemic Mo del with a General Nonlinear Saturated Incidence Rate in a Continuous Reactor

2019 ◽  
pp. 1-17
Author(s):  
Miled El Hajji ◽  
Sayed Sayari
Keyword(s):  
Fractional Order ◽  
Disease Transmission ◽  
System Modeling ◽  
Equilibrium Points ◽  
Continuous Reactor ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Dimensional Dynamical System

In this paper, I propose a fractional-order mathematical five-dimensional dynamical system  modeling a SVEIR model of infectious disease transmission in a chemostat. A profound qualitative analysis is given. The analysis of the local and global stability of equilibrium points is carried out. It is proved that if R > 1, then the disease-persistence (endemic) equilibrium is globally asymptotically stable. However, if R ≤ 1, then the disease-free equilibrium is globally asymptotically stable in R 5. Finally, some numerical tests are done using the ”PECE” method in order to validate the obtained results.

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