Modeling the Effect of Inpatient Rehabilitation of Tobacco Smokers on Smoking Dynamics

Aims: Develop and analyze a mathematical model of the effect of inpatient rehabilitation of tobacco smokers on tobacco smoking using Kenya as a case study. Perform stability analysis on the smoking free equilibrium point and endemic equilibrium point of the model. Use numerical simulation to investigate the impact of inpatient rehabilitation of tobacco smokers on smoking. Place and Duration of Study: Department of Mathematics and Actuarial Science, School of Pure and Applied Sciences (SPAS), Kenyatta University, Kenya, between May 2019 and September 2020. Tobacco smoking is a serious burden in Kenya and the world at large. Smoking harms nearly every organ of the body and affects the overall health of a person. Despite the overwhelming facts about the consequences of tobacco smoking, it remains a bad wont which is socially accepted and widely spread. In this research we numerically analyze the dynamics of smoking incorporating the impact of inpatient rehabilitation to curb the smoking habit. We first present a three-compartment model incorporating inpatient rehabilitation, then develop the system of ordinary differential equations governing the smoking dynamics. The basic reproduction number is determined using next generation matrix method. The model equilibria were computed and the stability analysis carried out. The results of stability analysis indicate that the disease-free equilibrium (DFE) is both locally and globally asymptotically stable for RS < 1 while the endemic equilibrium is both locally and globally asymptotically stable for RS > 1. Numerical simulations of model carried out with the help of MATLAB shows that, when rehabilitation is implemented effectively, it helps in minimization of smoking in the community.

Download Full-text

Mathematical Approach to Investigate Stress due to Control Measures to Curb COVID-19

Computational and Mathematical Methods in Medicine ◽

10.1155/2022/7772263 ◽

2022 ◽

Vol 2022 ◽

pp. 1-23

Author(s):

James Nicodemus Paul ◽

Silas Steven Mirau ◽

Isambi Sailon Mbalawata

Keyword(s):

Equilibrium Points ◽

Rank Correlation ◽

The Body ◽

Endemic Equilibrium ◽

Least Square ◽

Control Measures ◽

Model Parameters ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

The Impact

COVID-19 is a world pandemic that has affected and continues to affect the social lives of people. Due to its social and economic impact, different countries imposed preventive measures that are aimed at reducing the transmission of the disease. Such control measures include physical distancing, quarantine, hand-washing, travel and boarder restrictions, lockdown, and the use of hand sanitizers. Quarantine, out of the aforementioned control measures, is considered to be more stressful for people to manage. When people are stressed, their body immunity becomes weak, which leads to multiplying of coronavirus within the body. Therefore, a mathematical model consisting of six compartments, Susceptible-Exposed-Quarantine-Infectious-Hospitalized-Recovered (SEQIHR) was developed, aimed at showing the impact of stress on the transmission of COVID-19 disease. From the model formulated, the positivity, bounded region, existence, uniqueness of the solution, the model existence of free and endemic equilibrium points, and local and global stability were theoretically proved. The basic reproduction number ( R 0 ) was derived by using the next-generation matrix method, which shows that, when R 0 < 1 , the disease-free equilibrium is globally asymptotically stable whereas when R 0 > 1 the endemic equilibrium is globally asymptotically stable. Moreover, the Partial Rank Correlation Coefficient (PRCC) method was used to study the correlation between model parameters and R 0 . Numerically, the SEQIHR model was solved by using the Rung-Kutta fourth-order method, while the least square method was used for parameter identifiability. Furthermore, graphical presentation revealed that when the mental health of an individual is good, the body immunity becomes strong and hence minimizes the infection. Conclusively, the control parameters have a significant impact in reducing the transmission of COVID-19.

Download Full-text

Local Dynamics of an SVIR Epidemic Model with Logistic Growth

CAUCHY ◽

10.18860/ca.v6i3.9891 ◽

2020 ◽

Vol 6 (3) ◽

pp. 122-132

Author(s):

Joko Harianto ◽

Inda Puspita Sari

Keyword(s):

Stability Analysis ◽

Equilibrium Point ◽

Basic Reproduction Number ◽

Local Stability ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Logistic Growth ◽

Asymptotically Stable ◽

Local Stability Analysis ◽

The Basic Reproduction Number

Discussion of local stability analysis of SVIR models in this article is included in the scope of applied mathematics. The purpose of this discussion was to provide results of local stability analysis that had not been discussed in some articles related to the SVIR model. The SVIR models discussed in this article involve logistics growth in the vaccinated compartment. The results obtained, i.e. if the basic reproduction number less than one and m is positive, then there is one equilibrium point i.e. E0 is locally asymptotically stable. In the field of epidemiology, this means that the disease will disappear from the population. However, if the basic reproduction number more than one and b1 more than b, then there are two equilibrium points i.e. disease-free equilibrium point denoted by E0 and the endemic equilibrium point denoted by E1*. In this case the endemic equilibrium point E1* is locally asymptotically stable. In the field of epidemiology, this means that the disease will remain in the population. The numerical simulation supports these results.

Download Full-text

THE IMPACT OF MEDIA COVERAGE ON THE DYNAMICS OF INFECTIOUS DISEASE

International Journal of Biomathematics ◽

10.1142/s1793524508000023 ◽

2008 ◽

Vol 01 (01) ◽

pp. 65-74 ◽

Cited By ~ 95

Author(s):

YIPING LIU ◽

JING-AN CUI

Keyword(s):

Infectious Disease ◽

Media Coverage ◽

Compartment Model ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

The Impact ◽

Globally Stable ◽

Disease Free

In this paper, we give a compartment model to discuss the influence of media coverage to the spreading and controlling of infectious disease in a given region. The model exhibits two equilibria: a disease-free and a unique endemic equilibrium. Stability analysis of the models shows that the disease-free equilibrium is globally asymptotically stable if the reproduction number (ℝ0), which depends on parameters, is less than unity. But if ℝ0 > 1, it is shown that a unique endemic equilibrium appears, which is asymptotically stable. On a special case, the endemic equilibrium is globally stable. We discuss the role of media coverage on the spreading based on the theory results.

Download Full-text

Mathematical Modelling on Double Quarantine Process in the Spread and Stability of Covid-19

10.21203/rs.3.rs-40189/v1 ◽

2020 ◽

Author(s):

Jangyadatta Behera ◽

Aswin Kumar Rauta ◽

Yerra Shankar Rao ◽

Sairam Patnaik

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Analytical Result ◽

The Stability ◽

The Impact ◽

Disease Free

Abstract In this paper, a mathematical model is proposed on the spread and control of corona virus disease2019 (COVID19) to ascertain the impact of pre quarantine for suspected individuals having travel history ,immigrants and new born cases in the susceptible class following the lockdown or shutdown rules and adopted the post quarantine process for infected class. Set of nonlinear ordinary differential equations (ODEs) are generated and parameters like natural mortality rate, rate of COVID-19 induced death, rate of immigrants, rate of transmission and recovery rate are integrated in the scheme. A detailed analysis of this model is conducted analytically and numerically. The local and global stability of the disease is discussed mathematically with the help of Basic Reproduction Number. The ODEs are solved numerically with the help of Runge-Kutta 4th order method and graphs are drawn using MATLAB software to validate the analytical result with numerical simulation. It is found that both results are in good agreement with the results available in the existing literatures. The stability analysis is performed for both disease free equilibrium and endemic equilibrium points. The theorems based on Routh-Hurwitz criteria and Lyapunov function are proved .It is found that the system is locally asymptotically stable at disease free and endemic equilibrium points for basic reproduction number less than one and globally asymptotically stable for basic reproduction number greater than one. Finding of this study suggest that COVID-19 would remain pandemic with the progress of time but would be stable in the long-term if the pre and post quarantine policy for asymptomatic and symptomatic individuals are implemented effectively followed by social distancing, lockdown and containment.

Download Full-text

Mathematical Modeling and Analysis of Transmission Dynamics and Control of Schistosomiasis

Journal of Applied Mathematics ◽

10.1155/2021/6653796 ◽

2021 ◽

Vol 2021 ◽

pp. 1-20

Author(s):

Ebrima Kanyi ◽

Ayodeji Sunday Afolabi ◽

Nelson Owuor Onyango

Keyword(s):

Equilibrium Point ◽

Control Strategies ◽

Equilibrium Points ◽

Threshold Value ◽

Endemic Equilibrium ◽

Transmission Dynamics ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Modeling And Analysis ◽

Disease Free

This paper presents a mathematical model that describes the transmission dynamics of schistosomiasis for humans, snails, and the free living miracidia and cercariae. The model incorporates the treated compartment and a preventive factor due to water sanitation and hygiene (WASH) for the human subpopulation. A qualitative analysis was performed to examine the invariant regions, positivity of solutions, and disease equilibrium points together with their stabilities. The basic reproduction number, R 0 , is computed and used as a threshold value to determine the existence and stability of the equilibrium points. It is established that, under a specific condition, the disease-free equilibrium exists and there is a unique endemic equilibrium when R 0 > 1 . It is shown that the disease-free equilibrium point is both locally and globally asymptotically stable provided R 0 < 1 , and the unique endemic equilibrium point is locally asymptotically stable whenever R 0 > 1 using the concept of the Center Manifold Theory. A numerical simulation carried out showed that at R 0 = 1 , the model exhibits a forward bifurcation which, thus, validates the analytic results. Numerical analyses of the control strategies were performed and discussed. Further, a sensitivity analysis of R 0 was carried out to determine the contribution of the main parameters towards the die out of the disease. Finally, the effects that these parameters have on the infected humans were numerically examined, and the results indicated that combined application of treatment and WASH will be effective in eradicating schistosomiasis.

Download Full-text

On a Discrete SEIR Epidemic Model with Exposed Infectivity, Feedback Vaccination and Partial Delayed Re-Susceptibility

Mathematics ◽

10.3390/math9050520 ◽

2021 ◽

Vol 9 (5) ◽

pp. 520

Author(s):

Manuel De la Sen ◽

Santiago Alonso-Quesada ◽

Asier Ibeas

Keyword(s):

Equilibrium Point ◽

Epidemic Model ◽

Transmission Rate ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Control Gain ◽

Disease Free ◽

Partial Immunity

A new discrete Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model is proposed, and its properties of non-negativity and (both local and global) asymptotic stability of the solution sequence vector on the first orthant of the state-space are discussed. The calculation of the disease-free and the endemic equilibrium points is also performed. The model has the following main characteristics: (a) the exposed subpopulation is infective, as it is the infectious one, but their respective transmission rates may be distinct; (b) a feedback vaccination control law on the Susceptible is incorporated; and (c) the model is subject to delayed partial re-susceptibility in the sense that a partial immunity loss in the recovered individuals happens after a certain delay. In this way, a portion of formerly recovered individuals along a range of previous samples is incorporated again to the susceptible subpopulation. The rate of loss of partial immunity of the considered range of previous samples may be, in general, distinct for the various samples. It is found that the endemic equilibrium point is not reachable in the transmission rate range of values, which makes the disease-free one to be globally asymptotically stable. The critical transmission rate which confers to only one of the equilibrium points the property of being asymptotically stable (respectively below or beyond its value) is linked to the unity basic reproduction number and makes both equilibrium points to be coincident. In parallel, the endemic equilibrium point is reachable and globally asymptotically stable in the range for which the disease-free equilibrium point is unstable. It is also discussed the relevance of both the vaccination effort and the re-susceptibility level in the modification of the disease-free equilibrium point compared to its reached component values in their absence. The influences of the limit control gain and equilibrium re-susceptibility level in the reached endemic state are also explicitly made viewable for their interpretation from the endemic equilibrium components. Some simulation examples are tested and discussed by using disease parameterizations of COVID-19.

Download Full-text

Mathematical modeling of the impact of temperature variations and immigration on malaria prevalence in Nigeria

International Journal of Biomathematics ◽

10.1142/s1793524521500674 ◽

2021 ◽

pp. 2150067

Author(s):

Eunice E. Ukwajunor ◽

Eno E. E. Akarawak ◽

Israel Olutunji Abiala

Keyword(s):

Equilibrium Point ◽

Disease Transmission ◽

Deterministic Model ◽

Malaria Incidence ◽

Malaria Prevalence ◽

Endemic Equilibrium ◽

Transmission Dynamics ◽

Marginal Effect ◽

Asymptotically Stable ◽

The Impact

The study examines the population-level impact of temperature variability and immigration on malaria prevalence in Nigeria, using a novel deterministic model. The model incorporates disease transmission by immigrants into the community. In the absence of immigration, the model is shown to exhibit the phenomenon of backward bifurcation. The disease-free equilibrium of the autonomous version of the model was found to be locally asymptotically stable in the absence of infective immigrants. However, the model exhibits an endemic equilibrium point when the immigration parameter is greater than zero. The endemic equilibrium point is seen to be globally asymptotically stable in the absence of disease-induced mortality. Uncertainty and sensitivity analysis of the model, using parameter values and ranges relevant to malaria transmission dynamics in Nigeria, shows that the top three parameters that drive malaria prevalence (with respect to [Formula: see text]) are the mosquito natural death rate ([Formula: see text]), mosquito biting rate ([Formula: see text]) and the transmission rates between humans and mosquitoes ([Formula: see text]). Numerical simulations of the model show that in Nigeria, malaria burden increases with increasing mean monthly temperature in the range of 22–28[Formula: see text]. Thus, this study suggests that control strategies for malaria should be intensified during this period. It is further shown that the proportion of infective immigrants has marginal effect on the transmission dynamics of the disease. Therefore, the simulations suggest that a reduction in the fraction of infective immigrants, either exposed or infectious, would significantly reduce the malaria incidence in a population.

Download Full-text

Analysis of a virus-resistant HIV-1 model with behavior change in non-progressors

BIOMATH ◽

10.11145/j.biomath.2020.06.143 ◽

2020 ◽

Vol 9 (1) ◽

pp. 2006143

Author(s):

Musa Rabiu ◽

Robert Willie ◽

Nabendra Parumasur

Keyword(s):

Equilibrium Point ◽

Equilibrium Points ◽

Behavioural Change ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Reproduction Numbers ◽

Special Case ◽

Disease Free ◽

Hiv 1

We develop a virus-resistant HIV-1 mathematical model with behavioural change in HIV-1 resistant non-progressors. The model has both disease-free and endemic equilibrium points that are proved to be locally asymptotically stable depending on the value of the associated reproduction numbers. In both models, a non-linear Goh{Volterra Lyapunov function was used to prove that the endemic equilibrium point is globally asymptotically stable for special case while the method of Castillo-Chavez was used to prove the global asymptotic stability of the disease-free equilibrium point. In both the analytic and numerical results, this study shows that in the context of resistance to HIV/AIDS, total abstinence can also play an important role in protection against this notorious infectious disease.

Download Full-text

Stability analysis of a dynamical model of tuberculosis with incomplete treatment

Advances in Difference Equations ◽

10.1186/s13662-020-02950-0 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Ihsan Ullah ◽

Saeed Ahmad ◽

Qasem Al-Mdallal ◽

Zareen A. Khan ◽

Hasib Khan ◽

...

Keyword(s):

Disease Transmission ◽

Contact Rate ◽

Asymptotically Stable ◽

Treatment Rate ◽

Globally Asymptotically Stable ◽

Reproduction Ratio ◽

Efficient Treatment ◽

Effective Contact ◽

The Impact ◽

Incomplete Treatment

Abstract A simple deterministic epidemic model for tuberculosis is addressed in this article. The impact of effective contact rate, treatment rate, and incomplete treatment versus efficient treatment is investigated. We also analyze the asymptotic behavior, spread, and possible eradication of the TB infection. It is observed that the disease transmission dynamics is characterized by the basic reproduction ratio $\Re _{0}$ ℜ 0 ; if $\Re _{0}<1$ ℜ 0 < 1 , there is only a disease-free equilibrium which is both locally and globally asymptotically stable. Moreover, for $\Re _{0}>1$ ℜ 0 > 1 , a unique positive endemic equilibrium exists which is globally asymptotically stable. The global stability of the equilibria is shown via Lyapunov function. It is also obtained that incomplete treatment of TB causes increase in disease infection while efficient treatment results in a reduction in TB. Finally, for the estimated parameters, some numerical simulations are performed to verify the analytical results. These numerical results indicate that decrease in the effective contact rate λ and increase in the treatment rate γ play a significant role in the TB infection control.

Download Full-text

Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

Abstract and Applied Analysis ◽

10.1155/2013/916130 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11 ◽

Cited By ~ 3

Author(s):

Yakui Xue ◽

Tiantian Li

Keyword(s):

Incidence Rate ◽

Epidemic Model ◽

Threshold Value ◽

Endemic Equilibrium ◽

Logistic Growth ◽

Global Dynamics ◽

Asymptotically Stable ◽

Sir Epidemic Model ◽

Globally Asymptotically Stable ◽

Sir Epidemic

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for anyτ, we show that the disease-free equilibrium is globally asymptotically stable; whenR0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for anyτ=0; whenR0>1, the disease will persist. However, for anyτ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

Download Full-text