MODELING AND PARAMETER ESTIMATION OF TUBERCULOSIS WITH APPLICATION TO CAMEROON

2011 ◽  
Vol 21 (07) ◽  
pp. 1999-2015 ◽  
Author(s):  
SAMUEL BOWONG ◽  
JURGEN KURTHS
Keyword(s):  
Parameter Estimation ◽  
Statistical Data ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Numerical Study ◽  
Epidemiological Data ◽  
Unknown Parameters ◽  
Tuberculosis Model ◽  
Epidemiological Features ◽  
Disease Free

This paper deals with the problem of modeling and parameter estimation of a deterministic model of tuberculosis (abbreviated as TB for tubercle bacillus). We first propose and analyze a tuberculosis model without seasonality that incorporates the essential biological and epidemiological features of the disease. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with one or more stable endemic equilibria when the associated basic reproduction number is less than unity. The statistical data of new TB cases show seasonal fluctuations in many countries. Then, we extend the proposed TB model by incorporating seasonality. We propose a numerical study to estimate unknown parameters according to demographic and epidemiological data in Cameroon. Simulation results are in good accordance with the seasonal variation of the reported new cases of active TB in Cameroon.

scholarly journals A Stochastic TB Model for a Crowded Environment

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Sibaliwe Maku Vyambwera ◽  
Peter Witbooi
Keyword(s):  
Compartmental Model ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Stochastic Perturbation ◽  
Equation Model ◽  
Ordinary Differential Equation Model ◽  
Differential Equation Model ◽  
The Stability ◽  
Crowded Environments ◽  
Disease Free

We propose a stochastic compartmental model for the population dynamics of tuberculosis. The model is applicable to crowded environments such as for people in high density camps or in prisons. We start off with a known ordinary differential equation model, and we impose stochastic perturbation. We prove the existence and uniqueness of positive solutions of a stochastic model. We introduce an invariant generalizing the basic reproduction number and prove the stability of the disease-free equilibrium when it is below unity or slightly higher than unity and the perturbation is small. Our main theorem implies that the stochastic perturbation enhances stability of the disease-free equilibrium of the underlying deterministic model. Finally, we perform some simulations to illustrate the analytical findings and the utility of the model.

MODELING THE IMPACT OF VOLUNTARY TESTING AND TREATMENT ON TUBERCULOSIS TRANSMISSION DYNAMICS

2012 ◽  
Vol 05 (04) ◽  
pp. 1250029 ◽  
Author(s):  
S. MUSHAYABASA ◽  
C. P. BHUNU
Keyword(s):  
Deterministic Model ◽  
Reproduction Number ◽  
Positive Impact ◽  
Transmission Dynamics ◽  
Effective Control ◽  
Reproductive Number ◽  
Voluntary Testing ◽  
Manifold Theory ◽  
The Impact ◽  
Disease Free

A deterministic model for evaluating the impact of voluntary testing and treatment on the transmission dynamics of tuberculosis is formulated and analyzed. The epidemiological threshold, known as the reproduction number is derived and qualitatively used to investigate the existence and stability of the associated equilibrium of the model system. The disease-free equilibrium is shown to be locally-asymptotically stable when the reproductive number is less than unity, and unstable if this threshold parameter exceeds unity. It is shown, using the Centre Manifold theory, that the model undergoes the phenomenon of backward bifurcation where the stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction number is less than unity. The analysis of the reproduction number suggests that voluntary tuberculosis testing and treatment may lead to effective control of tuberculosis. Furthermore, numerical simulations support the fact that an increase voluntary tuberculosis testing and treatment have a positive impact in controlling the spread of tuberculosis in the community.

scholarly journals Mathematical Analysis of an Immune-structured Hikungunya Transmission Model

2019 ◽  
Vol 12 (4) ◽  
pp. 1533-1552
Author(s):  
Kambire Famane ◽  
Gouba Elisée ◽  
Tao Sadou ◽  
Blaise Some
Keyword(s):  
Numerical Simulation ◽  
Asymptotic Stability ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Transmission Model ◽  
Acquired Immunity ◽  
Local Asymptotic Stability ◽  
Well Posed ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, we have formulated a new deterministic model to describe the dynamics of the spread of chikunguya between humans and mosquitoes populations. This model takes into account the variation in mortality of humans and mosquitoes due to other causes than chikungunya disease, the decay of acquired immunity and the immune sytem boosting. From the analysis, itappears that the model is well posed from the mathematical and epidemiological standpoint. The existence of a single disease free equilibrium has been proved. An explicit formula, depending on the parameters of the model, has been obtained for the basic reproduction number R0 which is used in epidemiology. The local asymptotic stability of the disease free equilibrium has been proved. The numerical simulation of the model has confirmed the local asymptotic stability of the diseasefree equilbrium and the existence of endmic equilibrium. The varying effects of the immunity parameters has been analyzed numerically in order to provide better conditions for reducing the transmission of the disease.

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 stochastic population model of cholera disease

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Peter J. Witbooi ◽  
Grant E. Muller ◽  
Marshall B. Ongansie ◽  
Ibrahim H. I. Ahmed ◽  
Kazeem O. Okosun
Keyword(s):  
Population Model ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Stochastic Perturbation ◽  
Recent Outbreak ◽  
Environmental Reservoir ◽  
Stochastic Population Model ◽  
Disease Free ◽  
The Basic Reproduction Number ◽  
Extinction Theorem

<p style='text-indent:20px;'>A cholera population model with stochastic transmission and stochasticity on the environmental reservoir of the cholera bacteria is presented. It is shown that solutions are well-behaved. In comparison with the underlying deterministic model, the stochastic perturbation is shown to enhance stability of the disease-free equilibrium. The main extinction theorem is formulated in terms of an invariant which is a modification of the basic reproduction number of the underlying deterministic model. As an application, the model is calibrated as for a certain province of Nigeria. In particular, a recent outbreak (2019) in Nigeria is analysed and featured through simulations. Simulations include making forward projections in the form of confidence intervals. Also, the extinction theorem is illustrated through simulations.</p>

scholarly journals Modeling the impact of early interventions on the transmission dynamics of coronavirus infection

F1000Research ◽  
2021 ◽  
Vol 10 ◽  
pp. 518
Author(s):  
Christopher Saaha Bornaa ◽  
Baba Seidu ◽  
Yakubu Ibrahim Seini
Keyword(s):  
Death Rate ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Transmission Probability ◽  
Transmission Dynamics ◽  
Early Interventions ◽  
Coronavirus Infection ◽  
The Impact ◽  
Disease Free

A deterministic model is proposed to describe the transmission dynamics of coronavirus infection with early interventions. Epidemiological studies have employed modeling to unravel knowledge that transformed the lives of families, communities, nations and the entire globe. The study established the stability of both disease free and endemic equilibria. Stability occurs when the reproduction number, R0, is less than unity for both disease free and endemic equilibrium points. The global stability of the disease-free equilibrium point of the model is established whenever the basic reproduction number R0 is less than or equal to unity. The reproduction number is also shown to be directly related to the transmission probability (β), rate at which latently infected individuals join the infected class (δ) and rate of recruitment (Λ). It is inversely related to natural death rate (μ), rate of early treatment (τ1), rate of hospitalization of infected individuals (θ) and Covid-induced death rate (σ). The analytical results established are confirmed by numerical simulation of the model.

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.

Epidemic model formulation and analysis for diarrheal infections caused by salmonella

SIMULATION ◽  
2017 ◽  
Vol 93 (7) ◽  
pp. 543-552 ◽  
Author(s):  
Ojaswita Chaturvedi ◽  
Mandu Jeffrey ◽  
Edward Lungu ◽  
Shedden Masupe
Keyword(s):  
Parameter Estimation ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Rank Correlation ◽  
Latin Hypercube Sampling ◽  
Elementary Model ◽  
Partial Rank Correlation ◽  
Parameter Values ◽  
Working Out ◽  
Disease Free

Epidemic modeling can be used to gain better understanding of infectious diseases, such as diarrhea. In the presented research, a continuous mathematical model has been formulated for diarrhea caused by salmonella. This model has been analyzed and simulated to be established in a functioning form. Elementary model analysis, such as working out the disease-free state and basic reproduction number, has been done for this model. The basic reproduction number has been calculated using the next generation matrix method. Stability analysis of the model has been done using the Routh–Hurwitz method. Sensitivity analysis and parameter estimation have been completed for the system too using MATLAB packages that work on the Latin Hypercube Sampling and Partial Rank Correlation Coefficient methods. It was established that as long as R0 < 1, there will be no epidemic. Upon simulation using assumed parameter values, the results produced comprehended the epidemic theory and practical situations. The system was proven stable using the Routh–Hurwitz criterion and parameter estimation was successfully completed. Salmonella diarrhea has been successfully modeled and analyzed in this research. This model has been flexibly built and it can be integrated onto certain platforms to be used as a predictive system to prevent further infections of salmonella diarrhea.

scholarly journals Mathematical Analysis of a Malaria Model with Partial Immunity to Reinfection

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Li-Ming Cai ◽  
Abid Ali Lashari ◽  
Il Hyo Jung ◽  
Kazeem Oare Okosun ◽  
Young Il Seo
Keyword(s):  
Deterministic Model ◽  
Reproduction Number ◽  
Global Analysis ◽  
Backward Bifurcation ◽  
Mass Action ◽  
Transmission Dynamics ◽  
Global Dynamics ◽  
Reinfection Rate ◽  
Malaria Model ◽  
Disease Free

A deterministic model with variable human population for the transmission dynamics of malaria disease, which allows transmission by the recovered humans, is first developed and rigorously analyzed. The model reveals the presence of the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with one or more stable endemic equilibria when the associated reproduction number is less than unity. This phenomenon may arise due to the reinfection of host individuals who recovered from the disease. The model in an asymptotical constant population is also investigated. This results in a model with mass action incidence. A complete global analysis of the model with mass action incidence is given, which reveals that the global dynamics of malaria disease with reinfection is completely determined by the associated reproduction number. Moreover, it is shown that the phenomenon of backward bifurcation can be removed by replacing the standard incidence function with a mass action incidence. Graphical representations are provided to study the effect of reinfection rate and to qualitatively support the analytical results on the transmission dynamics of malaria.

scholarly journals Dynamics of Stochastically Perturbed SIS Epidemic Model with Vaccination

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yanan Zhao ◽  
Daqing Jiang
Keyword(s):  
Epidemic Model ◽  
Death Rate ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Dynamical Behavior ◽  
Standard Technique ◽  
Deterministic Models ◽  
Sis Epidemic Model ◽  
Disease Free ◽  
Sis Epidemic

We introduce stochasticity into an SIS epidemic model with vaccination. The stochasticity in the model is a standard technique in stochastic population modeling. In the deterministic models, the basic reproduction numberR0is a threshold which determines the persistence or extinction of the disease. When the perturbation and the disease-related death rate are small, we carry out a detailed analysis on the dynamical behavior of the stochastic model, also regarding of the value ofR0. IfR0≤1, the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model, whereas, ifR0>1, there is a stationary distribution, which means that the disease will prevail. The results are illustrated by computer simulations.

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