Drug-sensitivity and passive immunity mathematical epidemiological model for tuberculosis

Regardless of many decades of research, the widespread availability of a vaccine and more recently highly visible WHO efforts to promote a unified global control strategy, Tuberculosis remains a leading cause of infectious mortality. In this paper, a Mathematical Model for Tuberculosis Epidemic with Passive Immunity and Drug-Sensitivity is presented. We carried out analytical studies of the model where the population comprises of eight compartments: passively immune infants, susceptible, latently infected with DS-TB. The Disease Free Equilibrium (DFE) and the Endemic Equilibrium (EE) points were established. The next generation matrix method was used to obtain the reproduction number for drug sensitive (𝑅𝑜𝑠) Tuberculosis. We obtained the disease-free equilibrium for drug sensitive TB which is locally asymptotically stable when 𝑅𝑜𝑠 < 1 indicating that tuberculosis eradication is possible within the population. We also obtained the global stability of the disease-free equilibrium and results showed that the disease-free equilibrium point is globally asymptotically stable when 𝑅𝑜𝑠 ≤ 1 which indicates that tuberculosis naturally dies out.

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

Computational and Mathematical Methods in Medicine ◽

10.1155/2021/6664483 ◽

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.

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Stability of a Mathematical Model of Malaria Transmission with Relapse

Abstract and Applied Analysis ◽

10.1155/2014/289349 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9 ◽

Cited By ~ 5

Author(s):

Hai-Feng Huo ◽

Guang-Ming Qiu

Keyword(s):

Mathematical Model ◽

Basic Reproduction Number ◽

Recovery Rate ◽

Reproduction Number ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Next Generation Matrix ◽

The Government ◽

Disease Free ◽

The Basic Reproduction Number

A more realistic mathematical model of malaria is introduced, in which we not only consider the recovered humans return to the susceptible class, but also consider the recovered humans return to the infectious class. The basic reproduction numberR0is calculated by next generation matrix method. It is shown that the disease-free equilibrium is globally asymptotically stable ifR0≤1, and the system is uniformly persistence ifR0>1. Some numerical simulations are also given to explain our analytical results. Our results show that to control and eradicate the malaria, it is very necessary for the government to decrease the relapse rate and increase the recovery rate.

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Dynamical System of the Mathematical Model for Tuberculosis with Vaccination

ComTech Computer Mathematics and Engineering Applications ◽

10.21512/comtech.v10i2.5686 ◽

2019 ◽

Vol 10 (2) ◽

Author(s):

Dian Grace Ludji ◽

Paian Sianturi ◽

Endar Nugrahani

Keyword(s):

Dynamical System ◽

Equilibrium Point ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Prompt Treatment ◽

Latently Infected ◽

Next Generation Matrix ◽

The Mathematical Model ◽

Disease Free

This research focused on the modification of deterministic mathematical models for tuberculosis with vaccination. It also aimed to see the effect of giving the vaccine. It was done by adding vaccine compartments to people who were given the vaccine in the susceptible compartment. The population was divided into nine different groups. Those were susceptible individuals (S), vaccine (V), new latently infected (E1), diagnosed latently infected (E2), undiagnosed latently infected (E3), undiagnosed actively infected (l), diagnosed actively infected with prompt treatment (Dr), diagnosed actively infected with delay treatment (Dp), and treated (T). Basic reproduction number was constructed using next-generation matrix. Sensitivity analysis was also conducted. The results show that the model comprises two equilibriums: diseasefree equilibrium (T0) and endemic equilibrium (T*). It also shows that there is a relationship between R0 and two equilibriums. Moreover, the disease-free equilibrium point is asymptotically stable local when it is R0 < 1. Then, the disease-endemic equilibrium point is asymptotically stable local when it is R0 > 1. Furthermore, the parameters of β, ρ, and γ are the most important parameter.

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Mathematical Modelling of Human African Trypanosomiasis Using Control Measures

Computational and Mathematical Methods in Medicine ◽

10.1155/2018/5293568 ◽

2018 ◽

Vol 2018 ◽

pp. 1-13 ◽

Cited By ~ 4

Author(s):

Hamenyimana Emanuel Gervas ◽

Nicholas Kwasi-Do Ohene Opoku ◽

Shamsuddeen Ibrahim

Keyword(s):

Human African Trypanosomiasis ◽

Reproduction Number ◽

Tsetse Fly ◽

Control Measures ◽

Asymptotically Stable ◽

African Trypanosomiasis ◽

Globally Asymptotically Stable ◽

Vector Borne ◽

Next Generation Matrix ◽

Disease Free

Human African trypanosomiasis (HAT), commonly known as sleeping sickness, is a neglected tropical vector-borne disease caused by trypanosome protozoa. It is transmitted by bites of infected tsetse fly. In this paper, we first present the vector-host model which describes the general transmission dynamics of HAT. In the tsetse fly population, the HAT is modelled by three compartments, while in the human population, the HAT is modelled by four compartments. The next-generation matrix approach is used to derive the basic reproduction number, R0, and it is also proved that if R0≤1, the disease-free equilibrium is globally asymptotically stable, which means the disease dies out. The disease persists in the population if the value of R0>1. Furthermore, the optimal control model is determined by using the Pontryagin’s maximum principle, with control measures such as education, treatment, and insecticides used to optimize the objective function. The model simulations confirm that the use of the three control measures is very efficient and effective to eliminate HAT in Africa.

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Mathematical Analysis of TB Model with Vaccination and Saturated Incidence Rate

Abstract and Applied Analysis ◽

10.1155/2020/6669997 ◽

2020 ◽

Vol 2020 ◽

pp. 1-10

Author(s):

Ashenafi Kelemu Mengistu ◽

Peter J. Witbooi

Keyword(s):

Local Stability ◽

Reproduction Number ◽

Model System ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Latently Infected ◽

Saturated Incidence Rate ◽

Saturated Incidence ◽

Disease Free ◽

The Basic Reproduction Number

The model system of ordinary differential equations considers two classes of latently infected individuals, with different risk of becoming infectious. The system has positive solutions. By constructing a Lyapunov function, it is proved that if the basic reproduction number is less than unity, then the disease-free equilibrium point is globally asymptotically stable. The Routh-Hurwitz criterion is used to prove the local stability of the endemic equilibrium when R 0 > 1 . The model is illustrated using parameters applicable to Ethiopia. A variety of numerical simulations are carried out to illustrate our main results.

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Dynamical Analysis of an SEIRS Model With Incomplete Recovery and Relapse on Networks and Its Optimal Vaccination Control

10.21203/rs.3.rs-510785/v1 ◽

2021 ◽

Author(s):

Lei Zhang ◽

Maoxing Liu ◽

Qiang Hou ◽

Boli Xie

Keyword(s):

Control Strategy ◽

Reproduction Number ◽

Dynamical Analysis ◽

Global Dynamics ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Seirs Epidemic Model ◽

Theoretical Results ◽

Disease Free ◽

The Basic Reproduction Number

Abstract For some infectious diseases, such as herpes and tuberculosis, there is incomplete recovery and relapse. These phenomena make them difficult to control. In consequence of this status, an SEIRS epidemic model with incomplete recovery and relapse on networks is established and the global dynamics is studied. The results show that when the basic reproduction number R 0 <=1 the disease-free equilibrium is globally asymptotically stable; when R 0 > 1, the endemic equilibrium is globally asymptotically stable. In addition, in consideration of vaccination control strategy, an SVEIRS model is introduced and the optimal control is solved. At last, the theoretical results are illustrated with numerical simulations.

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Mathematical Model of Cholera Transmission with Education Campaign and Treatment Through Quarantine

Journal of Advances in Mathematics and Computer Science ◽

10.9734/jamcs/2019/v32i330145 ◽

2019 ◽

pp. 1-12

Author(s):

H. O. Nyaberi ◽

D. M. Malonza

Keyword(s):

Mathematical Model ◽

Health Education ◽

Reproduction Number ◽

Equilibrium Points ◽

Watery Diarrhea ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Education Campaign ◽

Next Generation Matrix ◽

Disease Free

Cholera, a water-borne disease characterized by intense watery diarrhea, affects people in the regions with poor hygiene and untreated drinking water. This disease remains a menace to public health globally and it indicates inequity and lack of community development. In this research, SIQR-B mathematical model based on a system of ordinary differential equations is formulated to study the dynamics of cholera transmission with health education campaign and treatmentthrough quarantine as controls against epidemic in Kenya. The effective basic reproduction number is computed using the next generation matrix method. The equilibrium points of the model are determined and their stability is analysed. Results of stability analysis show that the disease free equilibrium is both locally and globally asymptotically stable R0 < 1 while the endemic equilibrium is both locally and globally asymptotically stable R0 > 1. Numerical simulation carried out using MATLAB software shows that when health education campaign is efficient, the number of cholera infected individuals decreases faster, implying that health education campaign is vital in controlling the spread of cholera disease.

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Mathematical Modelling of Human African Trypanosomiasis Using Control Measures

10.20944/preprints201810.0561.v1 ◽

2018 ◽

Author(s):

Hamenyimana Emanuel Gervas ◽

Nicholas Kwasi-Do Ohene Opoku ◽

Shamsuddeen Ibrahim

Keyword(s):

Human Population ◽

Human African Trypanosomiasis ◽

Reproduction Number ◽

Tsetse Fly ◽

Control Measures ◽

Asymptotically Stable ◽

African Trypanosomiasis ◽

Globally Asymptotically Stable ◽

Next Generation Matrix ◽

Disease Free

Human African Trypanosomiasis (HAT) commonly known as sleeping sickness, is a neglected tropical vector borne disease caused by trypanosome protozoa. It is transmitted by bites of infected tsetse fly. In this paper we first present the vector-host model which describes the general transmission dynamics of HAT. In the tsetse fly population, the HAT is modelled by three compartments while in the human population, the HAT is modelled by four compartments. The next generation matrix approach is used to derive the basic reproduction number, R0, and also it is proved that if R0 ≤ 1 the disease free equilibrium is globally asymptotically stable, which means the disease dies out. The disease persist in the population if the value of R0 > 1. Furthermore, the optimal control model is determined by using the Pontryagin’s maximum principle with control measures such as education, treatment and insecticides used to optimize the objective function. The model simulations confirm that the use of the three control measures are very efficient and effective to eliminate HAT in Africa.

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

Bulletin of Mathematical Sciences and Applications ◽

10.18052/www.scipress.com/bmsa.9.53 ◽

2014 ◽

Vol 9 ◽

pp. 53-68 ◽

Cited By ~ 1

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

Mathematics ◽

10.3390/math6120328 ◽

2018 ◽

Vol 6 (12) ◽

pp. 328 ◽

Cited By ~ 3

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.

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