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.

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.

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.

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 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 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.

scholarly journals Dynamics of swine influenza model with optimal control

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Takasar Hussain ◽  
Muhammad Ozair ◽  
Kazeem Oare Okosun ◽  
Muhammad Ishfaq ◽  
Aziz Ullah Awan ◽  
...  
Keyword(s):  
Deterministic Model ◽  
Reproduction Number ◽  
Control Strategies ◽  
Influenza Pandemic ◽  
Swine Influenza ◽  
Transmission Dynamics ◽  
Model Parameters ◽  
And Control ◽  
Disease Free ◽  
Good Agreement

AbstractTransmission dynamics of swine influenza pandemic is analysed through a deterministic model. Qualitative analysis of the model includes global asymptotic stability of disease-free and endemic equilibria under a certain condition based on the reproduction number. Sensitivity analysis to ponder the effect of model parameters on the reproduction number is performed and control strategies are designed. It is also verified that the obtained numerical results are in good agreement with the analytical ones.

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.

scholarly journals Mathematical Model of Three Age-Structured Transmission Dynamics of Chikungunya Virus

2016 ◽  
Vol 2016 ◽  
pp. 1-31 ◽  
Author(s):  
Folashade B. Agusto ◽  
Shamise Easley ◽  
Kenneth Freeman ◽  
Madison Thomas
Keyword(s):  
Age Structure ◽  
Stable Disease ◽  
Chikungunya Virus ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Age Distribution ◽  
Transmission Dynamics ◽  
Globally Asymptotically Stable ◽  
Age Structured ◽  
Disease Free

We developed a new age-structured deterministic model for the transmission dynamics of chikungunya virus. The model is analyzed to gain insights into the qualitative features of its associated equilibria. Some of the theoretical and epidemiological findings indicate that the stable disease-free equilibrium is globally asymptotically stable when the associated reproduction number is less than unity. Furthermore, the model undergoes, in the presence of disease induced mortality, the phenomenon of backward bifurcation, where the stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number is less than unity. Further analysis of the model indicates that the qualitative dynamics of the model are not altered by the inclusion of age structure. This is further emphasized by the sensitivity analysis results, which shows that the dominant parameters of the model are not altered by the inclusion of age structure. However, the numerical simulations show the flaw of the exclusion of age in the transmission dynamics of chikungunya with regard to control implementations. The exclusion of age structure fails to show the age distribution needed for an effective age based control strategy, leading to a one size fits all blanket control for the entire population.

DYNAMICS ANALYSIS OF A VACCINATION MODEL FOR HPV TRANSMISSION

2014 ◽  
Vol 22 (04) ◽  
pp. 555-599 ◽  
Author(s):  
ALIYA A. ALSALEH ◽  
ABBA B. GUMEL
Keyword(s):  
Deterministic Model ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Dynamics Analysis ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Effective Community ◽  
Hpv Types ◽  
Special Case ◽  
Disease Free

A new deterministic model for the transmission dynamics of human papillomavirus (HPV) and related cancers, in the presence of the Gardasil vaccine (which targets four HPV types), is presented. In the absence of routine vaccination in the community, the model is shown to undergo the phenomenon of backward bifurcation. This phenomenon, which has important consequences on the feasibility of effective disease control in the community, arises due to the re-infection of recovered individuals. For the special case when backward bifurcation does not occur, the disease-free equilibrium (DFE) of the model is shown to be globally-asymptotically stable (GAS) if the associated reproduction number is less than unity. The model with vaccination is also rigorously analyzed. Numerical simulations of the model with vaccination show that, with the assumed 90% efficacy of the Gardasil vaccine, the effective community-wide control of the four Gardasil-preventable HPV types is feasible if the Gardasil coverage rate is high enough (in the range 78–88%).

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

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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