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.

scholarly journals A Mathematical Model of a Tuberculosis Transmission Dynamics Incorporating First and Second Line Treatment

2020 ◽  
Vol 24 (5) ◽  
pp. 917-922
Author(s):  
J. Andrawus ◽  
F.Y. Eguda ◽  
I.G. Usman ◽  
S.I. Maiwa ◽  
I.M. Dibal ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Equilibrium Point ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Line Treatment ◽  
Second Line ◽  
Tuberculosis Transmission ◽  
Disease Free

This paper presents a new mathematical model of a tuberculosis transmission dynamics incorporating first and second line treatment. We calculated a control reproduction number which plays a vital role in biomathematics. The model consists of two equilibrium points namely disease free equilibrium and endemic equilibrium point, it has been shown that the disease free equilibrium point was locally asymptotically stable if thecontrol reproduction number is less than one and also the endemic equilibrium point was locally asymptotically stable if the control reproduction number is greater than one. Numerical simulation was carried out which supported the analytical results. Keywords: Mathematical Model, Biomathematics, Reproduction Number, Disease Free Equilibrium, Endemic Equilibrium Point

scholarly journals MODELLING THE TRANSMISSION DYNAMICS OF CONTAGIOUS BOVINE PLEUROPNEUMONIA IN THE PRESENCE OF ANTIBIOTIC TREATMENT WITH LIMITED MEDICAL SUPPLY

2021 ◽  
Vol 26 (1) ◽  
pp. 1-20
Author(s):  
Achamyelesh A. Aligaz ◽  
Justin M. W. Munganga
Keyword(s):  
Antibiotic Treatment ◽  
Backward Bifurcation ◽  
Endemic Equilibrium ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Contagious Bovine Pleuropneumonia ◽  
Globally Asymptotically Stable ◽  
Medical Supply ◽  
Delayed Treatment ◽  
Disease Free

We present and analyze a mathematical model of the transmission dynamics of Contagious Bovine Pleuropneumonia (CBPP) in the presence of antibiotic treatment with limited medical supply. We use a saturated treatment function to model the effect of delayed treatment. We prove that there exist one disease free equilibrium and at most two endemic equilibrium solutions. A backward bifurcation occurs for small values of delay constant such that two endemic equilibriums exist if Rt (R*t,1); where, Rt is the treatment reproduction number and R*t is a threshold such that the disease dies out if and persists in the population if Rt > R*t. However, when a backward bifurcation occurs, a disease free system may easily be shifted to an epidemic. The bifurcation turns forward when the delay constant increases; thus, the disease free equilibrium becomes globally asymptotically stable if Rt < 1, and there exist unique and globally asymptotically stable endemic equilibrium if Rt > 1. However, the amount of maximal medical resource required to control the disease increases as the value of the delay constant increases. Thus, antibiotic treatment with limited medical supply setting would not successfully control CBPP unless we avoid any delayed treatment, improve the efficacy and availability of medical resources or it is given along with vaccination.

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.

Assessing the impact of transmissibility on a cluster-based COVID-19 model in India

2021 ◽  
pp. 2141002
Author(s):  
Tanvi ◽  
Mohammad Sajid ◽  
Rajiv Aggarwal ◽  
Ashutosh Rajput
Keyword(s):  
Reproduction Number ◽  
Transmission Rate ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Nonlinear Mathematical Model ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
The Stability ◽  
The Impact ◽  
Disease Free

In this paper, we have proposed a nonlinear mathematical model of different classes of individuals for coronavirus (COVID-19). The model incorporates the effect of transmission and treatment on the occurrence of new infections. For the model, the basic reproduction number [Formula: see text] has been computed. Corresponding to the threshold quantity [Formula: see text], the stability of endemic and disease-free equilibrium (DFE) points are determined. For [Formula: see text], if the endemic equilibrium point exists, then it is locally asymptotically stable, whereas the DFE point is globally asymptotically stable for [Formula: see text] which implies the eradication of the disease. The effects of various parameters on the spread of COVID-19 are discussed in the segment of sensitivity analysis. The model is numerically simulated to understand the effect of reproduction number on the transmission dynamics of the disease COVID-19. From the numerical simulations, it is concluded that if the reproduction number for the coronavirus disease is reduced below unity by decreasing the transmission rate and detecting more number of infectives, then the epidemic can be eradicated from the population.

scholarly journals Global Stability Analysis of SEIR Model with Holling Type II Incidence Function

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Mohammad A. Safi ◽  
Salisu M. Garba
Keyword(s):  
Deterministic Model ◽  
Reproduction Number ◽  
Communicable Disease ◽  
Endemic Equilibrium ◽  
Type Ii ◽  
Asymptotically Stable ◽  
Volterra Type ◽  
Globally Asymptotically Stable ◽  
Global Stability Analysis ◽  
Disease Free

A deterministic model for the transmission dynamics of a communicable disease is developed and rigorously analysed. The model, consisting of five mutually exclusive compartments representing the human dynamics, has a globally asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basicreproduction number(ℛ0), is less than unity; in such a case the endemic equilibrium does not exist. On the other hand, when the reproduction number is greater than unity, it is shown, using nonlinear Lyapunov function of Goh-Volterra type, in conjunction with the LaSalle's invariance principle, that the unique endemic equilibrium of the model is globally asymptotically stable under certain conditions. Furthermore, the disease is shown to be uniformly persistent wheneverℛ0>1.

Global stability of SEIVR epidemic model with generalized incidence and preventive vaccination

2015 ◽  
Vol 08 (06) ◽  
pp. 1550082 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Yasir Khan ◽  
Qaiser Badshah ◽  
Saeed Islam
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
Preventive Vaccination ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an SEIVR epidemic model with generalized incidence and preventive vaccination is considered. First, we formulate the model and obtain its basic properties. Then, we find the equilibrium points of the model, the disease-free and the endemic equilibrium. The stability of disease-free and endemic equilibrium is associated with the basic reproduction number [Formula: see text]. If the basic reproduction number [Formula: see text], the disease-free equilibrium is locally as well as globally asymptotically stable. Moreover, if the basic reproduction number [Formula: see text], the disease is uniformly persistent and the unique endemic equilibrium of the system is locally as well as globally asymptotically stable under certain conditions. Finally, the numerical results justify the analytical results.

The backward bifurcation of a model for malaria infection

2018 ◽  
Vol 11 (02) ◽  
pp. 1850018 ◽  
Author(s):  
Juan Wang ◽  
Xue-Zhi Li ◽  
Souvik Bhattacharya
Keyword(s):  
Malaria Infection ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Global Asymptotical Stability ◽  
Globally Asymptotically Stable ◽  
Vector Borne ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an epidemic model of a vector-borne disease, namely, malaria, is considered. The explicit expression of the basic reproduction number is obtained, the local and global asymptotical stability of the disease-free equilibrium is proved under certain conditions. It is shown that the model exhibits the phenomenon of backward bifurcation where the stable disease-free equilibrium coexists with a stable endemic equilibrium. Further, it is proved that the unique endemic equilibrium is globally asymptotically stable under certain conditions.

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%).

scholarly journals A Mathematical Model for Coinfection of Listeriosis and Anthrax Diseases

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Shaibu Osman ◽  
Oluwole Daniel Makinde
Keyword(s):  
Human Population ◽  
Reproduction Number ◽  
Zoonotic Diseases ◽  
Transmission Dynamics ◽  
Model Parameters ◽  
Contact Rate ◽  
Disease Dynamics ◽  
Human Contact ◽  
The Stability ◽  
Disease Free

Listeriosis and Anthrax are fatal zoonotic diseases caused by Listeria monocytogene and Bacillus Anthracis, respectively. In this paper, we proposed and analysed a compartmental Listeriosis-Anthrax coinfection model describing the transmission dynamics of Listeriosis and Anthrax epidemic in human population using the stability theory of differential equations. Our model revealed that the disease-free equilibrium of the Anthrax model only is locally stable when the basic reproduction number is less than one. Sensitivity analysis was carried out on the model parameters in order to determine their impact on the disease dynamics. Numerical simulation of the coinfection model was carried out and the results are displayed graphically and discussed. We simulate the Listeriosis-Anthrax coinfection model by varying the human contact rate to see its effects on infected Anthrax population, infected Listeriosis population, and Listeriosis-Anthrax coinfected population.

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.

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