scholarly journals Epidemic Model and Mathematical Study of Impact of Vaccination for the Control of Malware in Computer Network

2021 ◽  
pp. 72-96
Author(s):  
Titus Ifeanyi Chinebu ◽  
Ikechukwu Valentine Udegbe ◽  
Adanma Cecilia Eberendu
Keyword(s):  
Compartmental Model ◽  
Computer Network ◽  
Reproduction Number ◽  
Nonlinear Ordinary Differential Equation ◽  
Backward Bifurcation ◽  
Asymptotically Stable ◽  
Mathematical Study ◽  
Globally Asymptotically Stable ◽  
Bifurcation Phenomenon ◽  
Asymptotical Stable

Malware remains a significant threat to computer network.  In this paper, we consideredthe problem which computer malware cause to personal computers with its control by proposing a compartmental model SVEIRS (Susceptible Vaccinated-Exposed-infected-Recovered-Susceptible) for malware transmission in computer network using nonlinear ordinary differential equation. Through the analysis of the model, the basic reproduction number  were obtained, and the malware free equilibrium was proved to be locally asymptotical stable if  is less than unity and globally asymptotically stable if Ro is less than some threshold using a Lyapunov function. Also, the unique endemic equilibrium exists under certain conditions and the model underwent backward bifurcation phenomenon. To illustrate our theoretical analysis, some numerical simulation of the system was performed with RungeKutta fourth order (KR4) method in Mathlab. This was used in analyzing the behavior of different compartments of the model and the results showed that vaccination and treatment is very essential for malware control.

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 Mathematical analysis and optimal control interventions for sex structured syphilis model with three stages of infection and loss of immunity

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Abdulfatai Atte Momoh ◽  
Yusuf Bala ◽  
Dekera Jacob Washachi ◽  
Dione Déthié
Keyword(s):  
Reproduction Number ◽  
Nonlinear Ordinary Differential Equation ◽  
Backward Bifurcation ◽  
Asymptotically Stable ◽  
Infected Male ◽  
Globally Asymptotically Stable ◽  
Three Stages ◽  
Males And Females ◽  
The Stability ◽  
Stability Results

AbstractIn this study, we develop a nonlinear ordinary differential equation to study the dynamics of syphilis transmission incorporating controls, namely prevention and treatment of the infected males and females. We obtain syphilis-free equilibrium (SFE) and syphilis-present equilibrium (SPE). We obtain the basic reproduction number, which can be used to control the transmission of the disease, and thus establish the conditions for local and global stability of the syphilis-free equilibrium. The stability results show that the model is locally asymptotically stable if the Routh–Hurwitz criteria are satisfied and globally asymptotically stable. The bifurcation analysis result reveals that the model exhibits backward bifurcation. We adopted Pontryagin’s maximum principle to determine the optimality system for the syphilis model, which was solved numerically to show that syphilis transmission can be optimally best control using a combination of condoms usage and treatment in the primary stage of infection in both infected male and female populations.

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 Dynamics ofMycobacteriumandbovine tuberculosisin a Human-Buffalo Population

2014 ◽  
Vol 2014 ◽  
pp. 1-20 ◽  
Author(s):  
A. S. Hassan ◽  
S. M. Garba ◽  
A. B. Gumel ◽  
J. M.-S. Lubuma
Keyword(s):  
Mycobacterium Tuberculosis ◽  
Bovine Tuberculosis ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Sensitivity Analyses ◽  
Human Model ◽  
Cumulative Number ◽  
Globally Asymptotically Stable ◽  
Bifurcation Phenomenon ◽  
Using Data

A new model for the transmission dynamics ofMycobacterium tuberculosisand bovine tuberculosis in a community, consisting of humans and African buffalos, is presented. The buffalo-only component of the model exhibits the phenomenon of backward bifurcation, which arises due to the reinfection of exposed and recovered buffalos, when the associated reproduction number is less than unity. This model has a unique endemic equilibrium, which is globally asymptotically stable for a special case, when the reproduction number exceeds unity. Uncertainty and sensitivity analyses, using data relevant to the dynamics of the two diseases in the Kruger National Park, show that the distribution of the associated reproduction number is less than unity (hence, the diseases would not persist in the community). Crucial parameters that influence the dynamics of the two diseases are also identified. Both the buffalo-only and the buffalo-human model exhibit the same qualitative dynamics with respect to the local and global asymptotic stability of their respective disease-free equilibrium, as well as with respect to the backward bifurcation phenomenon. Numerical simulations of the buffalo-human model show that the cumulative number ofMycobacterium tuberculosiscases in humans (buffalos) decreases with increasing number of bovine tuberculosis infections in humans (buffalo).

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 Impact of Treatment on HIV-Malaria CoInfection Based on Mathematical Modeling

2019 ◽  
Vol 39 ◽  
pp. 45-62
Author(s):  
Amit Kumar Saha ◽  
Ashrafi Meher Niger ◽  
Chandra Nath Podder
Keyword(s):  
Treatment Strategy ◽  
Reproduction Number ◽  
Treatment Strategies ◽  
Backward Bifurcation ◽  
Asymptotically Stable ◽  
Single Treatment ◽  
Globally Asymptotically Stable ◽  
Mixed Treatment ◽  
The Impact ◽  
Disease Free

The distribution of HIV and malaria overlap globally. So there is always a chance of co-infection. In this paper the impact of medication on HIV-Malaria co-infection has been analyzed and we have developed a mathematical model using the idea of the models of Mukandavire, et al. [13] and Barley, et al. [3] where treatment classes are included. The disease-free equilibrium (DFE) of the HIV-only model is globally-asymptotically stable (GAS) when the reproduction number is less than one. But it is shown that in the malaria-only model, there is a coexistence of stable disease-free equilibrium and stable endemic equilibrium, for a certain interval of the reproduction number less than unity. This indicates the existence of backward bifurcation. Numerical simulations of the full model are performed to determine the impact of treatment strategies. It is shown that malaria-only treatment strategy reduces more new cases of the mixed infection than the HIV-only treatment strategy. Moreover, mixed treatment strategy reduces the least number of new cases compared to single treatment strategies. GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 45-62

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%).

NUMERICAL AND STABILITY ANALYSIS OF THE TRANSMISSION DYNAMICS OF SVIR EPIDEMIC MODEL WITH STANDARD INCIDENCE RATE

2019 ◽  
Vol 4 (2) ◽  
pp. 349 ◽  
Author(s):  
Oluwatayo Michael Ogunmiloro ◽  
Fatima Ohunene Abedo ◽  
Hammed Kareem
Keyword(s):  
Reproduction Number ◽  
Disease Spread ◽  
Asymptotically Stable ◽  
Equilibrium Solutions ◽  
Transform Method ◽  
Globally Asymptotically Stable ◽  
Differential Transform ◽  
Mathematical Techniques ◽  
Fourth Order Method ◽  
Theoretical Results

In this article, a Susceptible – Vaccinated – Infected – Recovered (SVIR) model is formulated and analysed using comprehensive mathematical techniques. The vaccination class is primarily considered as means of controlling the disease spread. The basic reproduction number (Ro) of the model is obtained, where it was shown that if Ro<1, at the model equilibrium solutions when infection is present and absent, the infection- free equilibrium is both locally and globally asymptotically stable. Also, if Ro>1, the endemic equilibrium solution is locally asymptotically stable. Furthermore, the analytical solution of the model was carried out using the Differential Transform Method (DTM) and Runge - Kutta fourth-order method. Numerical simulations were carried out to validate the theoretical results. 

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.

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