scholarly journals Mathematical Analysis of Transfusion – Transmitted Malaria Model with Optimal Control

2018 ◽  
Author(s):  
Michael Olaniyi Adeniyi ◽  
Oluwaseun Raphael Aderele
Keyword(s):  
Optimal Control ◽  
Research Work ◽  
Comparison Method ◽  
Asymptotically Stable ◽  
Control Pair ◽  
Perfect Agreement ◽  
Reproduction Numbers ◽  
Malaria Model ◽  
Disease Free ◽  
Blood Screening

An SIR (Susceptible – Infected – Removed) mathematical model for the transmission dynamics of the Transfusion –Transmitted Malaria (TTM) model with optimal control pair and was developed and studied in this research work. The model Transfusion –Transmitted Malaria disease – free equilibrium and endemic equilibriums points were determined. The model exhibited two equilibriums; disease-free and endemic equilibrium. It was shown that the disease – free equilibrium was locally asymptotically stable if the associated basic reproduction numbers  is less than unity while the disease persists if  is greater than unity. The global stability of the Transfusion –Transmitted Malaria model at the disease – free equilibrium was established using the comparison method. The optimality system was derived and an optimal control model of blood screening and drug treatment for the Transfusion –Transmitted Malaria model was investigated. Conditions for the optimal control were considered using Pontryagin’s Maximum Principle and solved numerically using the Forward and Backward Finite Difference Method (FBDM). Numerical results obtained are in perfect agreement with our analytical results.

scholarly journals Mathematical Analysis of Transfusion—Transmitted Malaria Model with Optimal Control

2019 ◽  
Author(s):  
Michael Olaniyi Adeniyi ◽  
Oluwaseun Raphael Aderele
Keyword(s):  
Optimal Control ◽  
Research Work ◽  
Comparison Method ◽  
Asymptotically Stable ◽  
Control Pair ◽  
Perfect Agreement ◽  
Reproduction Numbers ◽  
Malaria Model ◽  
Disease Free ◽  
Blood Screening

An SIRS (Susceptible–Infected–Removed-Susceptible) mathematical model for the transmission dynamics of the Transfusion–Transmitted Malaria (TTM) model with optimal control pair u1(t) and u2(t) was developed and studied in this research work. The model Transfusion–Transmitted Malaria disease–free equilibrium and endemic equilibriums points were determined. The model exhibited two equilibriums; disease-free and endemic equilibrium. It is shown that the disease–free equilibrium was locally asymptotically stable if the associated basic reproduction numbers R0 is less than unity while the disease persists if R0 is greater than unity. The global stability of the Transfusion–Transmitted Malaria model at the disease-free equilibrium was established using the comparison method. The optimality system was derived and an optimal control model of blood screening and drug treatment for the Transfusion–Transmitted Malaria model was investigated. Conditions for the optimal control were considered using Pontryagin’s Maximum Principle and solved numerically using the Forward and Backward Finite Difference Method (FBDM). Numerical results obtained are in perfect agreement with our analytical results.

Stability analysis and optimal control of an epidemic model with vaccination

2015 ◽  
Vol 08 (03) ◽  
pp. 1550030 ◽  
Author(s):  
Swarnali Sharma ◽  
G. P. Samanta
Keyword(s):  
Optimal Control ◽  
Stability Analysis ◽  
Epidemic Model ◽  
Asymptotically Stable ◽  
Control Pair ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
The Cost ◽  
Objective Functional ◽  
Disease Free

In this paper, we have developed a compartment of epidemic model with vaccination. We have divided the total population into five classes, namely susceptible, exposed, infective, infective in treatment and recovered class. We have discussed about basic properties of the system and found the basic reproduction number (R0) of the system. The stability analysis of the model shows that the system is locally as well as globally asymptotically stable at disease-free equilibrium E0when R0< 1. When R0> 1 endemic equilibrium E1exists and the system becomes locally asymptotically stable at E1under some conditions. We have also discussed the epidemic model with two controls, vaccination control and treatment control. An objective functional is considered which is based on a combination of minimizing the number of exposed and infective individuals and the cost of the vaccines and drugs dose. Then an optimal control pair is obtained which minimizes the objective functional. Our numerical findings are illustrated through computer simulations using MATLAB. Epidemiological implications of our analytical findings are addressed critically.

scholarly journals Optimal Control Model for the Transmission Dynamics of Malaria-Pneumonia Co-Infection with Mass Action Incidence

2018 ◽  
Author(s):  
Michael Olaniyi Adeniyi ◽  
Temitayo Olabisi Oluyo
Keyword(s):  
Optimal Control ◽  
Child Survival ◽  
Optimal Strategies ◽  
Comparison Method ◽  
Mass Action ◽  
Bed Nets ◽  
Infection Model ◽  
Reproduction Numbers ◽  
Infection Models ◽  
Malaria Model

Malaria and Pneumonia are leading causes of serious illness in children and adults worldwide with their death rate and prevalence on the rise. Such alarming statistics may retard the milestones so far achieved in meeting the Millennium Development Goals 4 and 6 whose targets are to improve child survival and reverse the high prevalence of diseases such as pneumonia and malaria respectively. Two sub-models of malaria-pneumonia co-infection namely malaria model and pneumonia model were considered first and then followed by the full malaria-pneumonia co-infection model. The malaria model, pneumonia model and co-infection model basic reproduction numbers denoted by Rm, Rp and Rmp respectively was obtained using the Next Generation Matrix method. The model disease free equilibrium&rsquo;s local and global stability was analysed using Descartes&rsquo; Rule of signs and Comparison method. The bifurcation analysis for the malaria, pneumonia and co-infection models was studied using the Centre Manifold Theory. The sensitivity indices of the model basic reproduction numbers Rm, Rp and Rmp to the parameters in the models were calculated. Optimal control theory was applied using the Pontryagins&rsquo; Maximum Principle to investigate optimal strategies for controlling the spread of malaria, pneumonia and co-infection models using insecticide treated bed nets (u1(t)) spraying of mosquitoes insecticides (u2(t)), sanitation (u3(t)), vaccination (u4(t)), anti-malaria drugs (u5(t)), anti-pneumonia drugs (u6(t)), both anti-malaria drugs and anti-pneumonia drugs (u7(t)) as the system time control variables.&nbsp; Numerical simulations using a set of parameter values were provided to validate the analytical results.

scholarly journals Modeling the effect of temperature variability on malaria control strategies

2020 ◽  
Vol 15 ◽  
pp. 65
Author(s):  
Salisu M. Garba ◽  
Usman A. Danbaba
Keyword(s):  
Control Strategies ◽  
Effect Of Temperature ◽  
Bed Nets ◽  
Asymptotically Stable ◽  
Temperature Changes ◽  
Offspring Number ◽  
Local Asymptotic Stability ◽  
Reproduction Numbers ◽  
The Impact ◽  
Disease Free

In this study, a non-autonomous (temperature dependent) and autonomous (temperature independent) models for the transmission dynamics of malaria in a population are designed and rigorously analysed. The models are used to assess the impact of temperature changes on various control strategies. The autonomous model is shown to exhibit the phenomenon of backward bifurcation, where an asymptotically-stable disease-free equilibrium (DFE) co-exists with an asymptotically-stable endemic equilibrium when the associated reproduction number is less than unity. This phenomenon is shown to arise due to the presence of imperfect vaccines and disease-induced mortality rate. Threshold quantities (such as the basic offspring number, vaccination and host type reproduction numbers) and their interpretations for the models are presented. Conditions for local asymptotic stability of the disease-free solutions are computed. Sensitivity analysis using temperature data obtained from Kwazulu Natal Province of South Africa [K. Okuneye and A.B. Gumel. Mathematical Biosciences 287 (2017) 72–92] is used to assess the parameters that have the most influence on malaria transmission. The effect of various control strategies (bed nets, adulticides and vaccination) were assessed via numerical simulations.

THE EFFECTS OF VACCINATION AND TREATMENT ON THE SPREAD OF HIV/AIDS

2004 ◽  
Vol 12 (04) ◽  
pp. 399-417 ◽  
Author(s):  
M. KGOSIMORE ◽  
E. M. LUNGU
Keyword(s):  
Equilibrium Point ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Treatment Programs ◽  
Asymptotically Stable ◽  
Reproduction Numbers ◽  
Threshold Conditions ◽  
Existence And Stability ◽  
Disease Free ◽  
Hiv Aids

This study investigates the effects of vaccination and treatment on the spread of HIV/AIDS. The objectives are (i) to derive conditions for the success of vaccination and treatment programs and (ii) to derive threshold conditions for the existence and stability of equilibria in terms of the effective reproduction number R. It is found, firstly, that the success of a vaccination and treatment program is achieved when R0t<R0, R0t<R0v and γeRVT(σ)<RUT(α), where R0t and R0v are respectively the reproduction numbers for populations consisting entirely of treated and vaccinated individuals, R0 is the basic reproduction number in the absence of any intervention, RUT(α) and RVT(σ) are respectively the reproduction numbers in the presence of a treatment (α) and a combination of vaccination and treatment (σ) strategies. Secondly, that if R<1, there exists a unique disease free equilibrium point which is locally asymptotically stable, while if R>1 there exists a unique locally asymptotically stable endemic equilibrium point, and that the two equilibrium points coalesce at R=1. Lastly, it is concluded heuristically that the stable disease free equilibrium point exists when the conditions R0t<R0, R0t<R0v and γeRVT(σ)<RUT(α) are satisfied.

scholarly journals Analysis of a Malaria Transmission Model in Children

2020 ◽  
Vol 24 (5) ◽  
pp. 789-798
Author(s):  
F.Y. Eguda ◽  
A.C. Ocheme ◽  
M.M. Sule ◽  
J. Andrawus ◽  
I.B. Babura
Keyword(s):  
Malaria Transmission ◽  
Compartmental Model ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Transmission Model ◽  
Acquired Immunity ◽  
Asymptotically Stable ◽  
Threshold Parameter ◽  
Malaria Model ◽  
Disease Free

In this paper, a nine compartmental model for malaria transmission in children was developed and a threshold parameter called control reproduction number which is known to be a vital threshold quantity in controlling the spread of malaria was derived. The model has a disease free equilibrium which is locally asymptotically stable if the control reproduction number is less than one and an endemic equilibrium point which is also locally asymptotically stable if the control reproduction number is greater than one. The model undergoes a backward bifurcation which is caused by loss of acquired immunity of recovered children and the rate at which exposed children progress to the mild stage of infection. Keywords: Malaria, Model, Backward Bifurcation, Local Stability.

scholarly journals Mathematical Analysis of Malaria-Schistosomiasis Coinfection Model

2016 ◽  
Vol 2016 ◽  
pp. 1-19 ◽  
Author(s):  
E. A. Bakare ◽  
C. R. Nwozo
Keyword(s):  
Numerical Simulations ◽  
Endemic Equilibrium ◽  
Model Parameters ◽  
Asymptotically Stable ◽  
Reproduction Numbers ◽  
Mathematical Techniques ◽  
Parameter Values ◽  
The Impact ◽  
Number Of Individuals ◽  
Disease Free

We formulated and analysed a mathematical model to explore the cointeraction between malaria and schistosomiasis. Qualitative and comprehensive mathematical techniques have been applied to analyse the model. The local stability of the disease-free and endemic equilibrium was analysed, respectively. However, the main theorem shows that if RMS<1, then the disease-free equilibrium is locally asymptotically stable and the phase will vanish out of the host and if RMS>1, a unique endemic equilibrium is also locally asymptotically stable and the disease persists at the endemic steady state. The impact of schistosomiasis and its treatment on malaria dynamics is also investigated. Numerical simulations using a set of reasonable parameter values show that the two epidemics coexist whenever their reproduction numbers exceed unity. Further, results of the full malaria-schistosomiasis model also suggest that an increase in the number of individuals infected with schistosomiasis in the presence of treatment results in a decrease in malaria cases. Sensitivity analysis was further carried out to investigate the influence of the model parameters on the transmission and spread of malaria-schistosomiasis coinfection. Numerical simulations were carried out to confirm our theoretical findings.

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.

Lyapunov function and global stability for a two-strain SEIR model with bilinear and non-monotone incidence

2019 ◽  
Vol 12 (02) ◽  
pp. 1950021 ◽  
Author(s):  
Dounia Bentaleb ◽  
Saida Amine
Keyword(s):  
Lyapunov Method ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Reproduction Numbers ◽  
The Status ◽  
Incidence Functions ◽  
Disease Free ◽  
Disease Parameters

In this paper, we study a multi-strain SEIR epidemic model with both bilinear and non-monotone incidence functions. Under biologically motivated assumptions, we show that the model has two basic reproduction numbers that we noted [Formula: see text] and [Formula: see text]; and four equilibrium points. Using the Lyapunov method, we prove that if [Formula: see text] and [Formula: see text] are less than one then the disease-free equilibrium is Globally Asymptotically Stable, thus the disease will be eradicated. However, if one of the two basic reproduction numbers is greater than one, then the strain that persists is that with the larger basic reproduction number. And finally if both of the two basic reproduction numbers are equal or greater than one then the total endemic equilibrium is globally asymptotically stable. A numerical simulation is also presented to illustrate the influence of the psychological effect, of people to infection, on the spread of the disease in the population. This simulation can be used to determine the status of different diseases in a region using the corresponding data and infectious disease parameters.

scholarly journals A Two-Strain Deterministic Dengue Model With Seasonality Effect

2019 ◽  
Vol 1 (2) ◽  
pp. 13-15
Author(s):  
Afeez Abidemi ◽  
Mohd Ismail Abd Aziz ◽  
Rohanin Ahmad
Keyword(s):  
Disease Transmission ◽  
Birth Rate ◽  
Climatic Factors ◽  
Secondary Infection ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Dengue Disease ◽  
Reproduction Numbers ◽  
Disease Free

A compartmental model is proposed to describe the dynamics of vector-host interations for dengue disease transmission with coexistence of two virus serotypes. The model is modified to incorporate  seasonal-dependent mosquito birth rate in order to examine the influence of climatic factors such as rainfall and temperature on the dynamics of mosquito population and dengue disease transmission. The Next Generation Matrix method is used to obtain the basic reproduction number associated with the model without seasonality effect. The global dynamics of the model is analysed using the Comparison Theorem. The model is simulated in MATLAB with ode45 routine for two cases, namely: the less aggressive case (Case ) and the more aggressive case (Case ).  Analysis of the model shows that the Disease-Free Equilibrium (DFE) is locally asymptotically stable whenever both the basic reproduction numbers  R01 (associated with strain  only) and  R0j (associated with strain  only) are below unity. It is shown that the DFE is globally asymptotically stable when the susceptibility indices for secondary infection in strain  1 ( sigma1) and strain j ( sigmaj), and  are all less than 1.

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