scholarly journals Mathematical Model for Optimal Control of Soil-Transmitted Helminth Infection

2020 ◽  
Vol 2020 ◽  
pp. 1-15 ◽  
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.

OPTIMAL CONTROL ANALYSIS OF THE EFFECT OF TREATMENT, ISOLATION AND VACCINATION ON HEPATITIS B VIRUS

2020 ◽  
Vol 28 (02) ◽  
pp. 351-376 ◽  
Author(s):  
MUHAMMAD ALTAF KHAN ◽  
SYED AZHAR ALI SHAH ◽  
SAIF ULLAH ◽  
KAZEEM OARE OKOSUN ◽  
MUHAMMAD FAROOQ
Keyword(s):  
Optimal Control ◽  
Hepatitis B ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Control Strategies ◽  
Control Measures ◽  
Hepatitis B Infection ◽  
Control Variables ◽  
Effect Of Treatment ◽  
The Basic Reproduction Number

Hepatitis B infection is a serious health issue and a major cause of deaths worldwide. This infection can be overcome by adopting proper treatment and control strategies. In this paper, we develop and use a mathematical model to explore the effect of treatment on the dynamics of hepatitis B infection. First, we formulate and use a model without control variables to calculate the basic reproduction number and to investigate basic properties of the model such as the existence and stability of equilibria. In the absence of control measures, we prove that the disease free equilibrium is locally asymptotically stable when the basic reproduction number is less than unity. Also, using persistent theorem, it is shown that the infection is uniformly persistent, whenever the basic reproduction number is greater than unity. Using optimal control theory, we incorporate into the model three time-dependent control variables and investigate the conditions required to curtail the spread of the disease. Finally, to illustrate the effectiveness of each of the control strategies on disease control and eradication, we perform numerical simulations. Based on the numerical results, we found that the first two strategies (treatment and isolation strategy) and (vaccination and isolation strategy) are not very effective as a long term control or eradication strategy for HBV. Hence, we recommend that in order to effectively control the disease, all the control measures (isolation, vaccination and treatment) must be implemented at the same time.

scholarly journals Estimating the basic reproduction number for the 2015 bubonic plague outbreak in Nyimba district of Eastern Zambia

2020 ◽  
Vol 14 (11) ◽  
pp. e0008811
Author(s):  
Joseph Sichone ◽  
Martin C. Simuunza ◽  
Bernard M. Hang’ombe ◽  
Mervis Kikonko
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Cost Effective ◽  
Control Measures ◽  
Effective Control ◽  
Infectious Period ◽  
Bubonic Plague ◽  
Infectious Individual ◽  
Susceptible Population ◽  
The Basic Reproduction Number

Background Plague is a re-emerging flea-borne infectious disease of global importance and in recent years, Zambia has periodically experienced increased incidence of outbreaks of this disease. However, there are currently no studies in the country that provide a quantitative assessment of the ability of the disease to spread during these outbreaks. This limits our understanding of the epidemiology of the disease especially for planning and implementing quantifiable and cost-effective control measures. To fill this gap, the basic reproduction number, R0, for bubonic plague was estimated in this study, using data from the 2015 Nyimba district outbreak, in the Eastern province of Zambia. R0 is the average number of secondary infections arising from a single infectious individual during their infectious period in an entirely susceptible population. Methodology/Principal findings Secondary epidemic data for the most recent 2015 Nyimba district bubonic plague outbreak in Zambia was analyzed. R0 was estimated as a function of the average epidemic doubling time based on the initial exponential growth rate of the outbreak and the average infectious period for bubonic plague. R0 was estimated to range between 1.5599 [95% CI: 1.382–1.7378] and 1.9332 [95% CI: 1.6366–2.2297], with average of 1.7465 [95% CI: 1.5093–1.9838]. Further, an SIR deterministic mathematical model was derived for this infection and this estimated R0 to be between 1.4 to 1.5, which was within the range estimated above. Conclusions/Significance This estimated R0 for bubonic plague is an indication that each bubonic plague case can typically give rise to almost two new cases during these outbreaks. This R0 estimate can now be used to quantitatively analyze and plan measurable interventions against future plague outbreaks in Zambia.

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

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.

scholarly journals The Role of Non-pharmacological Interventions on the Dynamics of Schistosomiasis

2021 ◽  
Vol 53 (2) ◽  
pp. 243-260
Author(s):  
Agatha Abokwara ◽  
Chinwendu Emilian Madubueze
Keyword(s):  
Public Health ◽  
Health Education ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Public Health Education ◽  
Control Measures ◽  
Snail Control ◽  
The Impact ◽  
Disease Free ◽  
The Basic Reproduction Number

Schistosomiasis is a neglected tropical disease affecting communities surrounded by water bodies where fishing activities take place or people go to swim, wash and cultivate crops. It poses a great risk to the health and economic life of inhabitants of the area. This study was carried out to evaluate the impact of public health education and snail control measures on the incidence of schistosomiasis. A model was developed with attention given to the snail and human populations that are the hosts of the cercariae and miracidia respectively. The existence and stability of disease-free and endemic equilibrium states were established. The disease-free and endemic equilibrium states were shown to be locally asymptotically stable whenever the basic reproduction number was less than unity. Numerical simulations of the model were carried out to evaluate the impact of interventions (public health education and snail control measures) on schistosomiasis transmission. It was observed that the implementation of low coverage snail control with highly efficacious molluscicide and massive public health education will make the basic reproduction number smaller than unity, which implies the eradication of schistosomiasis in the population.

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

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.

scholarly journals Optimal Control of a Dengue-Dengvaxia Model: Comparison Between Vaccination and Vector Control

2021 ◽  
Vol 9 (1) ◽  
pp. 198-212
Author(s):  
Cheryl Q. Mentuda
Keyword(s):  
Optimal Control ◽  
Vector Control ◽  
Basic Reproduction Number ◽  
Human Population ◽  
Model Comparison ◽  
Reproduction Number ◽  
Control Strategies ◽  
Minimum Principle ◽  
Transmitted Disease ◽  
The Basic Reproduction Number

Abstract Dengue is the most common mosquito-borne viral infection transmitted disease. It is due to the four types of viruses (DENV-1, DENV-2, DENV-3, DENV-4), which transmit through the bite of infected Aedes aegypti and Aedes albopictus female mosquitoes during the daytime. The first globally commercialized vaccine is Dengvaxia, also known as the CYD-TDV vaccine, manufactured by Sanofi Pasteur. This paper presents a Ross-type epidemic model to describe the vaccine interaction between humans and mosquitoes using an entomological mosquito growth population and constant human population. After establishing the basic reproduction number ℛ0, we present three control strategies: vaccination, vector control, and the combination of vaccination and vector control. We use Pontryagin’s minimum principle to characterize optimal control and apply numerical simulations to determine which strategies best suit each compartment. Results show that vector control requires shorter time applications in minimizing mosquito populations. Whereas vaccinating the primary susceptible human population requires a shorter time compared to the secondary susceptible human.

scholarly journals On an SEIR Epidemic Model with Vaccination of Newborns and Periodic Impulsive Vaccination with Eventual On-Line Adapted Vaccination Strategies to the Varying Levels of the Susceptible Subpopulation

2020 ◽  
Vol 10 (22) ◽  
pp. 8296 ◽  
Author(s):  
Malen Etxeberria-Etxaniz ◽  
Santiago Alonso-Quesada ◽  
Manuel De la Sen
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Disease Transmission ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Impulsive Vaccination ◽  
Disease Free ◽  
The Basic Reproduction Number

This paper investigates a susceptible-exposed-infectious-recovered (SEIR) epidemic model with demography under two vaccination effort strategies. Firstly, the model is investigated under vaccination of newborns, which is fact in a direct action on the recruitment level of the model. Secondly, it is investigated under a periodic impulsive vaccination on the susceptible in the sense that the vaccination impulses are concentrated in practice in very short time intervals around a set of impulsive time instants subject to constant inter-vaccination periods. Both strategies can be adapted, if desired, to the time-varying levels of susceptible in the sense that the control efforts be increased as those susceptible levels increase. The model is discussed in terms of suitable properties like the positivity of the solutions, the existence and allocation of equilibrium points, and stability concerns related to the values of the basic reproduction number. It is proven that the basic reproduction number lies below unity, so that the disease-free equilibrium point is asymptotically stable for larger values of the disease transmission rates under vaccination controls compared to the case of absence of vaccination. It is also proven that the endemic equilibrium point is not reachable if the disease-free one is stable and that the disease-free equilibrium point is unstable if the reproduction number exceeds unity while the endemic equilibrium point is stable. Several numerical results are investigated for both vaccination rules with the option of adapting through ime the corresponding efforts to the levels of susceptibility. Such simulation examples are performed under parameterizations related to the current SARS-COVID 19 pandemic.

scholarly journals Modeling the Effects of Helminth Infection on the Transmission Dynamics of Mycobacterium tuberculosis under Optimal Control Strategies

2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Aristide G. Lambura ◽  
Gasper G. Mwanga ◽  
Livingstone Luboobi ◽  
Dmitry Kuznetsov
Keyword(s):  
Optimal Control ◽  
Equilibrium Point ◽  
Drug Administration ◽  
Mass Drug Administration ◽  
Helminth Infection ◽  
Reproduction Number ◽  
Control Strategies ◽  
Active Tuberculosis ◽  
Control Measures ◽  
The Impact

A deterministic mathematical model for the transmission and control of cointeraction of helminths and tuberculosis is presented, to examine the impact of helminth on tuberculosis and the effect of control strategies. The equilibrium point is established, and the effective reproduction number is computed. The disease-free equilibrium point is confirmed to be asymptotically stable whenever the effective reproduction number is less than the unit. The analysis of the effective reproduction number indicates that an increase in the helminth cases increases the tuberculosis cases, suggesting that the control of helminth infection has a positive impact on controlling the dynamics of tuberculosis. The possibility of bifurcation is investigated using the Center Manifold Theorem. Sensitivity analysis is performed to determine the effect of every parameter on the spread of the two diseases. The model is extended to incorporate control measures, and Pontryagin’s Maximum Principle is applied to derive the necessary conditions for optimal control. The optimal control problem is solved numerically by the iterative scheme by considering vaccination of infants for Mtb, treatment of individuals with active tuberculosis, mass drug administration with regular antihelminthic drugs, and sanitation control strategies. The results show that a combination of educational campaign, treatment of individuals with active tuberculosis, mass drug administration, and sanitation is the most effective strategy to control helminth-Mtb coinfection. Thus, to effectively control the helminth-Mtb coinfection, we suggest to public health stakeholders to apply intervention strategies that are aimed at controlling helminth infection and the combination of vaccination of infants and treatment of individuals with active tuberculosis.

scholarly journals A prototype vaccination model for endemic Covid-19 under waning immunity and imperfect vaccine take-up

2021 ◽  
Author(s):  
John S Dagpunar ◽  
ChenChen Wu
Keyword(s):  
Basic Reproduction Number ◽  
Vaccine Efficacy ◽  
Reproduction Number ◽  
Vaccine Coverage ◽  
Infection Prevalence ◽  
State Variables ◽  
Susceptible Population ◽  
Waning Immunity ◽  
The Impact ◽  
The Basic Reproduction Number

In this paper, for an infectious disease such as Covid-19, we present a SIR model which examines the impact of waning immunity, vaccination rates, vaccine efficacy, and the proportion of the susceptible population who aspire to be vaccinated. Under an assumed constant control reproduction number, we provide simple conditions for the disease to be eliminated, and conversely for it to exhibit the more likely endemic behaviour. With regard to Covid-19, it is shown that if the control reproduction number is set to the basic reproduction number (say 6) of the dominant delta (B1.617.2) variant, vaccination alone, even under the most optimistic of assumptions about vaccine efficacy and high vaccine coverage, is very unlikely to lead to elimination of the disease. The model is not intended to be predictive but more an aid to understanding the relative importance of various biological and control parameters. For example, from a long-term perspective, it may be found that in the UK, through changes in societal behaviour (such as mask use, ventilation, and level of homeworking), without formal government interventions such as on-off lockdowns, the control reproduction number can still be maintained at a level significantly below the basic reproduction number. Even so, our simulations show that endemic behaviour ensues. The model obtains equilibrium values of the state variables such as the infection prevalence and mortality rate under various scenarios.

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