Optimal control of intervention strategies in malaria–tuberculosis co-infection with relapse

2018 ◽  
Vol 11 (02) ◽  
pp. 1850017 ◽  
Author(s):  
Peter Mpasho Mwamtobe ◽  
Simphiwe Mpumelelo Simelane ◽  
Shirley Abelman ◽  
Jean Michel Tchuenche
Keyword(s):  
Optimal Control ◽  
Sensitivity Analysis ◽  
Stable Disease ◽  
Disease Transmission ◽  
Backward Bifurcation ◽  
Intervention Strategies ◽  
Relative Importance ◽  
Disease Dynamics ◽  
Epidemiological Features ◽  
Disease Free

A model which incorporates some of the basic epidemiological features of the co-dynamics of malaria and tuberculosis (TB) is formulated and the effectiveness of current intervention strategies of these two diseases is analyzed. The malaria-only and TB-only models are considered first. Global stability disease-free steady states of the two sub-models does not hold due to the co-existence of stable disease-free with stable endemic equilibria, a phenomenon known as backward bifurcation. The dynamics of the dual malaria–TB model with intervention strategies are also analyzed. Numerical simulations of the malaria–TB model are carried out to determine whether the two diseases can co-exist. Lastly, sensitivity analysis on key parameters that drive the disease dynamics is performed in order to identify their relative importance to disease transmission.

OPTIMAL CONTROL OF THE SPREAD OF MALARIA SUPERINFECTIVITY

2013 ◽  
Vol 21 (04) ◽  
pp. 1340002 ◽  
Author(s):  
FOLASHADE AGUSTO ◽  
SUZANNE LENHART
Keyword(s):  
Optimal Control ◽  
Qualitative Analysis ◽  
Stable Disease ◽  
Optimal Control Theory ◽  
Deterministic Model ◽  
Backward Bifurcation ◽  
Time Dependent ◽  
Implementation Cost ◽  
Life Threatening ◽  
Disease Free

Malaria is a life-threatening disease caused by parasites that are transmitted to people through the bites of infected mosquitoes. In this paper, a deterministic model for malaria transmission, that incorporates superinfection is presented. Qualitative analysis of the model reveals the presence of backward bifurcation in which a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity. Optimal control theory is then applied to the model to study time-dependent treatment efforts to minimize the infected in individuals while keeping the implementation cost at a minimum.

scholarly journals Assessing the Potential Impact of Immunity Waning on the Dynamics of COVID-19: An Endemic Model of COVID-19

2021 ◽  
Author(s):  
MUSA RABIU ◽  
Sarafa A. Iyaniwura
Keyword(s):  
Disease Transmission ◽  
Reproduction Number ◽  
Transmission Rate ◽  
Backward Bifurcation ◽  
Disease Dynamics ◽  
Potential Impact ◽  
The Impact ◽  
Induced Immunity ◽  
Disease Free ◽  
Pharmaceutical Interventions

Abstract We developed an endemic model of COVID-19 to assess the impact of vaccination and immunity waning on the dynamics of the disease. Our model exhibits the phenomenon of backward bifurcation and bi-stability, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium. The epidemiological implication of this is that the control reproduction number being less than unity is no longer sufficient to guarantee disease eradication. We showed that this phenomenon could be eliminated by either increasing the vaccine efficacy or by reducing the disease transmission rate (adhering to non-pharmaceutical interventions). Furthermore, we numerically investigated the impacts of vaccination and waning of both vaccine-induced immunity and post-recovery immunity on the disease dynamics. Our simulation results show that the waning of vaccine-induced immunity has more effect on the disease dynamics relative to post-recovery immunity waning, and suggests that more emphasis should be on reducing the waning of vaccine-induced immunity to eradicate COVID-19.

A mathematical analysis of a tuberculosis epidemic model with two treatments and exogenous re-infection

2020 ◽  
pp. 2050082
Author(s):  
Mehdi Lotfi ◽  
Azizeh Jabbari ◽  
Hossein Kheiri
Keyword(s):  
Mathematical Model ◽  
Asymptotic Stability ◽  
Stable Disease ◽  
Reproduction Number ◽  
Geometric Approach ◽  
Backward Bifurcation ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Bifurcation Phenomenon ◽  
Disease Free

In this paper, we propose a mathematical model of tuberculosis with two treatments and exogenous re-infection, in which the treatment is effective for a number of infectious individuals and it fails for some other infectious individuals who are being treated. We show that the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibria when the related basic reproduction number is less than unity. Also, it is shown that under certain conditions the model cannot exhibit backward bifurcation. Furthermore, it is shown in the absence of re-infection, the backward bifurcation phenomenon does not exist, in which the disease-free equilibrium of the model is globally asymptotically stable when the associated reproduction number is less than unity. The global asymptotic stability of the endemic equilibrium, when the associated reproduction number is greater than unity, is established using the geometric approach. Numerical simulations are presented to illustrate our main results.

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.

CASE DETECTION AND DIRECT OBSERVATION THERAPY STRATEGY (DOTS) IN NIGERIA: ITS EFFECT ON TB DYNAMICS

2008 ◽  
Vol 16 (01) ◽  
pp. 1-31 ◽  
Author(s):  
DANIEL OKUONGHAE ◽  
VINCENT AIHIE
Keyword(s):  
Stable Disease ◽  
Latent Class ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Treatment Success ◽  
Backward Bifurcation ◽  
Case Detection ◽  
Detection Rates ◽  
Therapy Strategy ◽  
Disease Free

This paper addresses the synergy between case detection and the implementation of DOTS in Nigeria in the control of tuberculosis using a deterministic model which incorporates many of the essential biological and epidemiological features of TB as well as DOTS surveillance and implementation parameters for Nigeria. The model differentiated between individuals who progress to the "primary" latent stage when they got infected for the first time and those who progress to the "secondary" latent class depending on whether they failed treatment or due to self-cure. The model was shown to have a locally asymptotically stable disease free equilibrium where the reproduction number was less than unity. However, it was also shown that the model is capable of exhibiting the backward bifurcation phenomenon, where the stable disease free equilibrium co-exists with a stable endemic equilibrium where the reproduction number is less than unity. We saw that increasing the case detection parameter actually reduces the backward bifurcation range. For smaller exogenous re-infection values, increasing the case detection parameter could totally eliminate the bifurcation range. Uncertainty and sensitivity analysis using the Latin hypercube sampling technique was also carried out on the parameters as well as the reproduction number and the results showed that there were three parameters that were highly influential in determining the magnitude of the reproduction number; of the three, only one, the case detection parameter, was highly influential in reducing the magnitude of the reproduction number. Results from the numerical simulation and qualitative analysis showed that DOTS expansion in Nigeria must include significant increase in case detection rates, otherwise the impressive cure rates under DOTS will pale into insignificance with the rise in the number of undetected infectious persons and the number of "secondary" latent cases. Overall, the study shows that increasing the case detection rate will not only lower the backward bifurcation range, in the presence of exogenous re-infection, but could also lower the reproduction number, reducing the severity of the TB epidemic. This is possible as far as the current impressive treatment success rates under DOTS in Nigeria is sustained.

scholarly journals A co-infection model for Two-Strain Malaria and Cholera with Optimal Control

2020 ◽  
Author(s):  
Kenneth Uzoma Egeonu ◽  
Simeon Chioma Inyama ◽  
Andrew Omame
Keyword(s):  
Optimal Control ◽  
Drug Resistance ◽  
Necessary Conditions ◽  
Reproduction Number ◽  
Dynamical Property ◽  
Backward Bifurcation ◽  
Infection Model ◽  
Existence Of Optimal Control ◽  
The Impact ◽  
Disease Free

A mathematical model for two strains of Malaria and Cholera with optimal control is studied and analyzed to assess the impact of treatment controls in reducing the burden of the diseases in a population, in the presence of malaria drug resistance. The model is shown to exhibit the dynamical property of backward bifurcation when the associated reproduction number is less than unity. The global asymptotic stability of the disease-free equilibrium of the model is proven not to exist. The necessary conditions for the existence of optimal control and the optimality system for the model is established using the Pontryagin's Maximum Principle. Numerical simulations of the optimal control model reveal that malaria drug resistance can greatly influence the co-infection cases averted, even in the presence of treatment controls for co-infected individuals.

scholarly journals An SEIARD epidemic model for COVID-19 in Mexico: mathematical analysis and state-level forecast

2020 ◽  
Author(s):  
Ugo Avila-Ponce de León ◽  
Ángel G. C. Pérez ◽  
Eric Avila-Vales
Keyword(s):  
Disease Transmission ◽  
Local Stability ◽  
Reproduction Number ◽  
State Level ◽  
Model Parameters ◽  
Relative Importance ◽  
Current Outbreak ◽  
Sensibility Analysis ◽  
Next Generation Matrix ◽  
Disease Free

We propose an SEIARD mathematical model to investigate the current outbreak of coronavirus disease (COVID-19) in Mexico. Our model incorporates the asymptomatic infected individuals, who represent the majority of the infected population (with symptoms or not) and could play an important role in spreading the virus without any knowledge. We calculate the basic reproduction number (R0) via the next-generation matrix method and estimate the per day infection, death and recovery rates. The local stability of the disease free equilibrium is established in terms of R0. A sensibility analysis is performed to determine the relative importance of the model parameters to the disease transmission. We calibrate the parameters of the SEIARD model to the reported number of infected cases and fatalities for several states in Mexico by minimizing the sum of squared errors and attempt to forecast the evolution of the outbreak until August 2020.

Analysis of a mathematical model for the transmission dynamics of human melioidosis

2020 ◽  
Vol 13 (07) ◽  
pp. 2050062
Author(s):  
Yibeltal Adane Terefe ◽  
Semu Mitiku Kassa
Keyword(s):  
Human Population ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Endemic Equilibrium ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Disease Dynamics ◽  
Number Formula ◽  
Disease Free

A deterministic model for the transmission dynamics of melioidosis disease in human population is designed and analyzed. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the basic reproduction number [Formula: see text] is less than one. It is further shown that the backward bifurcation dynamics is caused by the reinfection of individuals who recovered from the disease and relapse. The existence of backward bifurcation implies that bringing down [Formula: see text] to less than unity is not enough for disease eradication. In the absence of backward bifurcation, the global asymptotic stability of the disease-free equilibrium is shown whenever [Formula: see text]. For [Formula: see text], the existence of at least one locally asymptotically stable endemic equilibrium is shown. Sensitivity analysis of the model, using the parameters relevant to the transmission dynamics of the melioidosis disease, is discussed. Numerical experiments are presented to support the theoretical analysis of the model. In the numerical experimentations, it has been observed that screening and treating individuals in the exposed class has a significant impact on the disease dynamics.

TRANSMISSION DYNAMICS AND OPTIMAL CONTROL OF AN INFLUENZA MODEL WITH QUARANTINE AND TREATMENT

2012 ◽  
Vol 05 (03) ◽  
pp. 1260011 ◽  
Author(s):  
WEI-WEI SHI ◽  
YUAN-SHUN TAN
Keyword(s):  
Optimal Control ◽  
Disease Transmission ◽  
Reproduction Number ◽  
Influenza Pandemic ◽  
Control Measures ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
Parameter Values ◽  
The Impact ◽  
Disease Free

We develop an influenza pandemic model with quarantine and treatment, and analyze the dynamics of the model. Analytical results of the model show that, if basic reproduction number [Formula: see text], the disease-free equilibrium (DFE) is globally asymptotically stable, if [Formula: see text], the disease is uniformly persistent. The model is then extended to assess the impact of three anti-influenza control measures, precaution, quarantine and treatment, by re-formulating the model as an optimal control problem. We focus primarily on controlling disease with a possible minimal the systemic cost. Pontryagin's maximum principle is used to characterize the optimal levels of the three controls. Numerical simulations of the optimality system, using a set of reasonable parameter values, indicate that the precaution measure is more effective in reducing disease transmission than the other two control measures. The precaution measure should be emphasized.

scholarly journals Mathematical Analysis of a Cholera Model with Vaccination

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Jing'an Cui ◽  
Zhanmin Wu ◽  
Xueyong Zhou
Keyword(s):  
Disease Transmission ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Disease Spread ◽  
Relative Importance ◽  
Globally Asymptotically Stable ◽  
Perform Sensitivity Analysis ◽  
Compound Matrix ◽  
Disease Free

We consider aSVR-Bcholera model with imperfect vaccination. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is established. We calculate the certain threshold known as the control reproduction numberℛv. Ifℛv<1, we obtain sufficient conditions for the global asymptotic stability of the disease-free equilibrium; the diseases will be eliminated from the community. By comparison of arguments, it is proved that ifℛv>1, the disease persists and the unique endemic equilibrium is globally asymptotically stable, which is obtained by the second compound matrix techniques and autonomous convergence theorems. We perform sensitivity analysis ofℛvon the parameters in order to determine their relative importance to disease transmission and show that an imperfect vaccine is always beneficial in reducing disease spread within the community.

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