Mathematical Study of a Class of Epidemiological Models with Multiple Infectious Stages

2020 ◽  
Vol 21 (3-4) ◽  
pp. 259-274
Author(s):  
S. Bowong ◽  
A. Temgoua ◽  
Y. Malong ◽  
J. Mbang
Keyword(s):  
Theoretical Study ◽  
Reproduction Number ◽  
Communicable Disease ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Epidemiological Models ◽  
Mathematical Study ◽  
Globally Asymptotically Stable ◽  
Disease Free

AbstractThis paper deals with the mathematical analysis of a general class of epidemiological models with multiple infectious stages for the transmission dynamics of a communicable disease. We provide a theoretical study of the model. We derive the basic reproduction number $\mathcal R_0$ that determines the extinction and the persistence of the infection. We show that the disease-free equilibrium is globally asymptotically stable whenever $\mathcal R_0 \leq 1$, while when $\mathcal R_0 \gt 1$, the disease-free equilibrium is unstable and there exists a unique endemic equilibrium point which is globally asymptotically stable. A case study for tuberculosis (TB) is considered to numerically support the analytical results.

scholarly journals A Mathematical Model of COVID-19 Pandemic: A Case Study of Bangkok, Thailand

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Pakwan Riyapan ◽  
Sherif Eneye Shuaib ◽  
Arthit Intarasit
Keyword(s):  
Mathematical Model ◽  
Reproduction Number ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Proposed Model ◽  
Next Generation Matrix ◽  
Disease Free ◽  
The Basic Reproduction Number

In this study, we propose a new mathematical model and analyze it to understand the transmission dynamics of the COVID-19 pandemic in Bangkok, Thailand. It is divided into seven compartmental classes, namely, susceptible S , exposed E , symptomatically infected I s , asymptomatically infected I a , quarantined Q , recovered R , and death D , respectively. The next-generation matrix approach was used to compute the basic reproduction number denoted as R cvd 19 of the proposed model. The results show that the disease-free equilibrium is globally asymptotically stable if R cvd 19 < 1 . On the other hand, the global asymptotic stability of the endemic equilibrium occurs if R cvd 19 > 1 . The mathematical analysis of the model is supported using numerical simulations. Moreover, the model’s analysis and numerical results prove that the consistent use of face masks would go on a long way in reducing the COVID-19 pandemic.

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.

scholarly journals Mathematical Analysis of COVID-19 Transmission Dynamics with a Case Study of Nigeria and its Computer Simulation

2020 ◽  
Author(s):  
B. C. Agbata ◽  
Ogala Emmanuel ◽  
Tenuche Bashir ◽  
Obeng-Denteh William
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Transmission Dynamics ◽  
Government Policies ◽  
Asymptotically Stable ◽  
Deadly Disease ◽  
Next Generation Matrix ◽  
Disease Free ◽  
The Basic Reproduction Number

AbstractIn this article, we formulated a mathematical model for the spread of the COVID-19 disease and we introduced quarantined and isolated compartments. The next generation matrix method was adopted to compute the basic reproduction number (R0) in order to assess the transmission dynamics of the COVID-19 deadly disease. Stability analysis of the disease free equilibrium is investigated based on the basic reproduction number and the result shows that it is locally and asymptotically stable for R0 less than 1. Numerical calculation of the basic reproduction number revealed that R0 < 1 which means that the disease can be eradicated from Nigeria. The study shows that isolation, quarantine and other government policies like social distancing and lockdown are the best approaches to control the pernicious nature of COVID-19 pandemic.

scholarly journals Mathematical Modelling of the Transmission Dynamics of Contagious Bovine Pleuropneumonia with Vaccination and Antibiotic Treatment

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Achamyelesh Amare Aligaz ◽  
Justin Manango W. Munganga
Keyword(s):  
Mathematical Model ◽  
Antibiotic Treatment ◽  
Reproduction Number ◽  
Threshold Value ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Contagious Bovine Pleuropneumonia ◽  
Sharp Threshold ◽  
Globally Asymptotically Stable ◽  
Disease Free

In this paper we present a mathematical model for the transmission dynamics of Contagious Bovine Pleuropneumonia (CBPP) by considering antibiotic treatment and vaccination. The model is comprised of susceptible, vaccinated, exposed, infectious, persistently infected, and recovered compartments. We analyse the model by deriving a formula for the control reproduction number Rc and prove that, for Rc<1, the disease free equilibrium is globally asymptotically stable; thus CBPP dies out, whereas for Rc>1, the unique endemic equilibrium is globally asymptotically stable and hence the disease persists. Thus, Rc=1 acts as a sharp threshold between the disease dying out or causing an epidemic. As a result, the threshold of antibiotic treatment is αt⁎=0.1049. Thus, without using vaccination, more than 85.45% of the infectious cattle should receive antibiotic treatment or the period of infection should be reduced to less than 8.15 days to control the disease. Similarly, the threshold of vaccination is ρ⁎=0.0084. Therefore, we have to vaccinate at least 80% of susceptible cattle in less than 49.5 days, to control the disease. Using both vaccination and antibiotic treatment, the threshold value of vaccination depends on the rate of antibiotic treatment, αt, and is denoted by ραt. Hence, if 50% of infectious cattle receive antibiotic treatment, then at least 50% of susceptible cattle should get vaccination in less than 73.8 days in order to control the disease.

scholarly journals A Delay Differential Equation Model of HIV Infection, with Therapy and CTL Response

2014 ◽  
Vol 9 ◽  
pp. 53-68 ◽  
Author(s):  
B. El Boukari ◽  
N. Yousfi
Keyword(s):  
Reproduction Number ◽  
Equation Model ◽  
Asymptotically Stable ◽  
Differential Equation Model ◽  
Globally Asymptotically Stable ◽  
Intracellular Delay ◽  
Immunodeficiency Virus ◽  
Delay Differential ◽  
Theoretical Results ◽  
Disease Free

In this work we investigate a new mathematical model that describes the interactions betweenCD4+ T cells, human immunodeficiency virus (HIV), immune response and therapy with two drugs.Also an intracellular delay is incorporated into the model to express the lag between the time thevirus contacts a target cell and the time the cell becomes actively infected. The model dynamicsis completely defined by the basic reproduction number R0. If R0 ≤ 1 the disease-free equilibriumis globally asymptotically stable, and if R0 > 1, two endemic steady states exist, and their localstability depends on value of R0. We show that the intracellular delay affects on value of R0 becausea larger intracellular delay can reduce the value of R0 to below one. Finally, numerical simulationsare presented to illustrate our theoretical results.

scholarly journals Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 328 ◽  
Author(s):  
Yanli Ma ◽  
Jia-Bao Liu ◽  
Haixia Li
Keyword(s):  
Lyapunov Function ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R 0 > 1 . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R 0 > 1 . Finally, some numerical simulations are presented to illustrate the analysis results.

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

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.

scholarly journals Seasonality Impact on the Transmission Dynamics of Tuberculosis

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Yali Yang ◽  
Chenping Guo ◽  
Luju Liu ◽  
Tianhua Zhang ◽  
Weiping Liu
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Autonomous Systems ◽  
Positive Periodic Solution ◽  
Transmission Dynamics ◽  
Uniform Persistence ◽  
Globally Asymptotically Stable ◽  
The Impact ◽  
Disease Free ◽  
The Basic Reproduction Number

The statistical data of monthly pulmonary tuberculosis (TB) incidence cases from January 2004 to December 2012 show the seasonality fluctuations in Shaanxi of China. A seasonality TB epidemic model with periodic varying contact rate, reactivation rate, and disease-induced death rate is proposed to explore the impact of seasonality on the transmission dynamics of TB. Simulations show that the basic reproduction number of time-averaged autonomous systems may underestimate or overestimate infection risks in some cases, which may be up to the value of period. The basic reproduction number of the seasonality model is appropriately given, which determines the extinction and uniform persistence of TB disease. If it is less than one, then the disease-free equilibrium is globally asymptotically stable; if it is greater than one, the system at least has a positive periodic solution and the disease will persist. Moreover, numerical simulations demonstrate these theorem results.

Sign in / Sign up

Export Citation Format

Share Document