Optimal Control Analysis of Pneumonia and Meningitis Coinfection

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.

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

10.1101/2020.07.23.20160473 ◽

2020 ◽

Cited By ~ 1

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.

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Optimal Control Analysis of a Cholera Epidemic Model

Biophysical Reviews and Letters ◽

10.1142/s1793048019500024 ◽

2019 ◽

Vol 14 (01) ◽

pp. 27-48 ◽

Cited By ~ 5

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.

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Analysis of the Model on the Effect of Seasonal Factors on Malaria Transmission Dynamics

Journal of Applied Mathematics ◽

10.1155/2020/8885558 ◽

2020 ◽

Vol 2020 ◽

pp. 1-19

Author(s):

Victor Yiga ◽

Hasifa Nampala ◽

Julius Tumwiine

Keyword(s):

Malaria Transmission ◽

Basic Reproduction Number ◽

Deterministic Model ◽

Reproduction Number ◽

Equilibrium Points ◽

Current Control ◽

Transmission Dynamics ◽

Mosquito Vector ◽

The Impact ◽

The Basic Reproduction Number

Malaria is one of the world’s most prevalent epidemics. Current control and eradication efforts are being frustrated by rapid changes in climatic factors such as temperature and rainfall. This study is aimed at assessing the impact of temperature and rainfall abundance on the intensity of malaria transmission. A human host-mosquito vector deterministic model which incorporates temperature and rainfall dependent parameters is formulated. The model is analysed for steady states and their stability. The basic reproduction number is obtained using the next-generation method. It was established that the mosquito population depends on a threshold value θ , defined as the number of mosquitoes produced by a female Anopheles mosquito throughout its lifetime, which is governed by temperature and rainfall. The conditions for the stability of the equilibrium points are investigated, and it is shown that there exists a unique endemic equilibrium which is locally and globally asymptotically stable whenever the basic reproduction number exceeds unity. Numerical simulations show that both temperature and rainfall affect the transmission dynamics of malaria; however, temperature has more influence.

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Mathematical modeling and optimal intervention strategies of the COVID-19 outbreak

10.21203/rs.3.rs-872671/v1 ◽

2021 ◽

Author(s):

Jayanta Mondal ◽

Subhas Khajanchi

Keyword(s):

Optimal Control ◽

Basic Reproduction Number ◽

Disease Transmission ◽

Numerical Solutions ◽

Reproduction Number ◽

Equilibrium Points ◽

Intervention Strategy ◽

Intervention Strategies ◽

Qualitative Behavior ◽

The Basic Reproduction Number

Abstract 32,737,939 active cases and 438,210 deaths because of COVID-19 pandemic were recorded on 30 August 2021 in India. To end this ongoing global COVID-19 pandemic, there is an urgent need to implement multiple population-wide policies like social distancing, testing more people and contact tracing. To predict the course of the pandemic and come up with a strategy to control it effectively, a compartmental model has been established. The following six stages of infection are taken into consideration: susceptible ($S$), asymptomatic infected ($A$), clinically ill or symptomatic infected ($I$), quarantine ($Q$), isolation ($J$) and recovered ($R$), collectively termed as SAIQJR. The qualitative behavior of the model and the stability of biologically realistic equilibrium points are investigated in terms of the basic reproduction number. We performed sensitivity analysis with respect to the basic reproduction number and obtained that the disease transmission rate has an impact in mitigating the spread of diseases. Moreover, considering the non-pharmaceutical and pharmaceutical intervention strategies as control functions, an optimal control problem is implemented to mitigate the disease fatality. To reduce the infected individuals and to minimize the cost of the controls, an objective functional has been constructed and solved with the aid of Pontryagin's Maximum Principle. The implementation of optimal control strategy at the start of a pandemic tends to decrease the intensity of epidemic peaks, spreading the maximal impact of an epidemic over an extended time period. Extensive numerical simulations show that the implementation of intervention strategy has an impact in controlling the transmission dynamics of COVID-19 epidemic. Further, our numerical solutions exhibit that the combination of three controls are more influential when compared with the combination of two controls as well as single control. Therefore the implementation of all the three control strategies may help to mitigate novel coronavirus disease transmission at this present epidemic scenario.

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

Advances in Difference Equations ◽

10.1186/s13662-021-03546-y ◽

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.

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Mathematical Model for Optimal Control of Soil-Transmitted Helminth Infection

Computational and Mathematical Methods in Medicine ◽

10.1155/2020/6721919 ◽

2020 ◽

Vol 2020 ◽

pp. 1-15 ◽

Cited By ~ 2

Author(s):

Aristide G. Lambura ◽

Gasper G. Mwanga ◽

Livingstone Luboobi ◽

Dmitry Kuznetsov

Keyword(s):

Optimal Control ◽

Health Education ◽

Basic Reproduction Number ◽

Helminth Infection ◽

Reproduction Number ◽

Equilibrium Points ◽

Control Measures ◽

Personal Hygiene ◽

Susceptible Population ◽

The Basic Reproduction Number

In this paper, we study the dynamics of soil-transmitted helminth infection. We formulate and analyse a deterministic compartmental model using nonlinear differential equations. The basic reproduction number is obtained and both disease-free and endemic equilibrium points are shown to be asymptotically stable under given threshold conditions. The model may exhibit backward bifurcation for some parameter values, and the sensitivity indices of the basic reproduction number with respect to the parameters are determined. We extend the model to include control measures for eradication of the infection from the community. Pontryagian’s maximum principle is used to formulate the optimal control problem using three control strategies, namely, health education through provision of educational materials, educational messages to improve the awareness of the susceptible population, and treatment by mass drug administration that target the entire population(preschool- and school-aged children) and sanitation through provision of clean water and personal hygiene. Numerical simulations were done using MATLAB and graphical results are displayed. The cost effectiveness of the control measures were done using incremental cost-effective ratio, and results reveal that the combination of health education and sanitation is the best strategy to combat the helminth infection. Therefore, in order to completely eradicate soil-transmitted helminths, we advise investment efforts on health education and sanitation controls.

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Backward Bifurcation in a Cholera Model: A Case Study of Outbreak in Zimbabwe and Haiti

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741750170x ◽

2017 ◽

Vol 27 (11) ◽

pp. 1750170

Author(s):

Sandeep Sharma ◽

Nitu Kumari

Keyword(s):

Basic Reproduction Number ◽

Deterministic Model ◽

Reproduction Number ◽

Numerical Study ◽

Equilibrium Points ◽

Geometric Approach ◽

Backward Bifurcation ◽

Global Dynamics ◽

Proposed Model ◽

The Basic Reproduction Number

In this paper, a nonlinear deterministic model is proposed with a saturated treatment function. The expression of the basic reproduction number for the proposed model was obtained. The global dynamics of the proposed model was studied using the basic reproduction number and theory of dynamical systems. It is observed that proposed model exhibits backward bifurcation as multiple endemic equilibrium points exist when [Formula: see text]. The existence of backward bifurcation implies that making [Formula: see text] is not enough for disease eradication. This, in turn, makes it difficult to control the spread of cholera in the community. We also obtain a unique endemic equilibria when [Formula: see text]. The global stability of unique endemic equilibria is performed using the geometric approach. An extensive numerical study is performed to support our analytical results. Finally, we investigate two major cholera outbreaks, Zimbabwe (2008–09) and Haiti (2010), with the help of the present study.

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Analysis of yellow fever prevention strategy from the perspective of mathematical model and cost-effectiveness analysis

Mathematical Biosciences and Engineering ◽

10.3934/mbe.2022084 ◽

2021 ◽

Vol 19 (2) ◽

pp. 1786-1824

Author(s):

Bevina D. Handari ◽

◽

Dipo Aldila ◽

Bunga O. Dewi ◽

Hanna Rosuliyana ◽

...

Keyword(s):

Mathematical Model ◽

Optimal Control ◽

Yellow Fever ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Equilibrium Points ◽

Intervention Strategies ◽

Yellow Fever Vaccine ◽

The Stability ◽

The Pontryagin Maximum Principle

<abstract><p>We developed a new mathematical model for yellow fever under three types of intervention strategies: vaccination, hospitalization, and fumigation. Additionally, the side effects of the yellow fever vaccine were also considered in our model. To analyze the best intervention strategies, we constructed our model as an optimal control model. The stability of the equilibrium points and basic reproduction number of the model are presented. Our model indicates that when yellow fever becomes endemic or disappears from the population, it depends on the value of the basic reproduction number, whether it larger or smaller than one. Using the Pontryagin maximum principle, we characterized our optimal control problem. From numerical experiments, we show that the optimal levels of each control must be justified, depending on the strategies chosen to optimally control the spread of yellow fever.</p></abstract>

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Parameter-dependent transmission dynamics and optimal control of foot and mouth disease in a contaminated environment

Journal of the Egyptian Mathematical Society ◽

10.1186/s42787-019-0058-1 ◽

2019 ◽

Vol 27 (1) ◽

Cited By ~ 1

Author(s):

Francis Mugabi ◽

Joseph Mugisha ◽

Betty Nannyonga ◽

Henry Kasumba ◽

Margaret Tusiime

Keyword(s):

Optimal Control ◽

Deterministic Model ◽

Foot And Mouth Disease ◽

Decay Rates ◽

Mouth Disease ◽

Control Measures ◽

Transmission Dynamics ◽

Optimality System ◽

Foot And Mouth ◽

Environmental Decontamination

AbstractThe problem of foot and mouth disease (FMD) is of serious concern to the livestock sector in most nations, especially in developing countries. This paper presents the formulation and analysis of a deterministic model for the transmission dynamics of FMD through a contaminated environment. It is shown that the key parameters that drive the transmission of FMD in a contaminated environment are the shedding, transmission, and decay rates of the virus. Using numerical results, it is depicted that the host-to-host route is more severe than the environmental-to-host route. The model is then transformed into an optimal control problem. Using the Pontryagin’s Maximum Principle, the optimality system is determined. Utilizing a gradient type algorithm with projection, the optimality system is solved for three control strategies: optimal use of vaccination, environmental decontamination, and a combination of vaccination and environmental decontamination. Results show that a combination of vaccination and environmental decontamination is the most optimal strategy. These results indicate that if vaccination and environmental decontamination are used optimally during an outbreak, then FMD transmission can be controlled. Future studies focusing on the control measures for the transmission of FMD in a contaminated environment should aim at reducing the transmission and the shedding rates, while increasing the decay rate.

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Mathematical analysis of a measles transmission dynamics model in Bangladesh with double dose vaccination

Scientific Reports ◽

10.1038/s41598-021-95913-8 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Md Abdul Kuddus ◽

M. Mohiuddin ◽

Azizur Rahman

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Transmission Rate ◽

Equilibrium Points ◽

Rank Correlation ◽

Dynamical Analysis ◽

Model Parameters ◽

Progression Rate ◽

Double Dose ◽

The Basic Reproduction Number

AbstractAlthough the availability of the measles vaccine, it is still epidemic in many countries globally, including Bangladesh. Eradication of measles needs to keep the basic reproduction number less than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{R}}_{0}<1)$$ ( i . e . R 0 < 1 ) . This paper investigates a modified (SVEIR) measles compartmental model with double dose vaccination in Bangladesh to simulate the measles prevalence. We perform a dynamical analysis of the resulting system and find that the model contains two equilibrium points: a disease-free equilibrium and an endemic equilibrium. The disease will be died out if the basic reproduction number is less than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{ R}}_{0}<1)$$ ( i . e . R 0 < 1 ) , and if greater than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{R}}_{0}>1)$$ ( i . e . R 0 > 1 ) epidemic occurs. While using the Routh-Hurwitz criteria, the equilibria are found to be locally asymptotically stable under the former condition on $${\mathrm{R}}_{0}$$ R 0 . The partial rank correlation coefficients (PRCCs), a global sensitivity analysis method is used to compute $${\mathrm{R}}_{0}$$ R 0 and measles prevalence $$\left({\mathrm{I}}^{*}\right)$$ I ∗ with respect to the estimated and fitted model parameters. We found that the transmission rate $$(\upbeta )$$ ( β ) had the most significant influence on measles prevalence. Numerical simulations were carried out to commissions our analytical outcomes. These findings show that how progression rate, transmission rate and double dose vaccination rate affect the dynamics of measles prevalence. The information that we generate from this study may help government and public health professionals in making strategies to deal with the omissions of a measles outbreak and thus control and prevent an epidemic in Bangladesh.

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