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

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.

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Mathematical Modelling and Analysis of Corruption of Morals amongst Adolescents with Control Measures in Kenya

Discrete Dynamics in Nature and Society ◽

10.1155/2021/6662185 ◽

2021 ◽

Vol 2021 ◽

pp. 1-16

Author(s):

Nathan Oigo Mokaya ◽

Haileyesus Tessema Alemmeh ◽

Cyrus Gitonga Ngari ◽

Grace Gakii Muthuri

Keyword(s):

Integrated Control ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Control Measures ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Age Appropriate ◽

Manifold Theory ◽

Necessary And Sufficient ◽

Well Posed

In the present paper, we formulate a new mathematical model for the dynamics of moral corruption with comprehensive age-appropriate sexual information and provision of guidance and counselling. The population is subdivided into three (3) different compartments according to their level of information on sexual matters. The model is proved to be both epidemiologically and mathematically well posed. The existence of unique morally corrupt-free and endemic equilibrium points is investigated. The basic reproduction number with respect to morally corrupt-free equilibrium is obtained using next generation matrix approach to monitor the dynamics of corrupt morals and ascertain its level in order to suggest effective intervention strategies to control this problem. The local as well as global asymptotic stability of these equilibrium points is studied. The analysis reveals a globally asymptotically stable morally corrupt-free equilibrium whenever ℛ 0 ≤ 1 and a globally asymptotically stable endemic equilibrium if otherwise. Further analysis, using center manifold theory, shows that the model exhibits forward bifurcation insinuating that the classical epidemiological requirement of ℛ 0 ≤ 1 is necessary and sufficient for elimination of moral corruption. A brief discussion on the graphical results using the available numerical procedures is shown. From numerical simulations, it was ascertain that integrated control strategy is the best approach to fight against moral corruption transmission. Lastly, some key parameters that show significance in the moral corruption elimination from the society are also exploited.

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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.

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

Journal of Advances in Mathematics and Computer Science ◽

10.9734/jamcs/2021/v36i430352 ◽

2021 ◽

pp. 1-14

Author(s):

Amos Mwendwa Nyamai ◽

Winifred N. Mutuku

Keyword(s):

Stability Analysis ◽

Equilibrium Point ◽

Tobacco Smoking ◽

Smoking Habit ◽

Inpatient Rehabilitation ◽

The Body ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

The Impact

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.

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Global Stability of Malaria Transmission Dynamics Model with Logistic Growth

Discrete Dynamics in Nature and Society ◽

10.1155/2018/5759834 ◽

2018 ◽

Vol 2018 ◽

pp. 1-12 ◽

Cited By ~ 1

Author(s):

Abadi Abay Gebremeskel

Keyword(s):

Sensitivity Analysis ◽

Global Stability ◽

Malaria Transmission ◽

Equilibrium Points ◽

Transmission Dynamics ◽

Logistic Growth ◽

Asymptotically Stable ◽

Dynamics Model ◽

Globally Asymptotically Stable ◽

The Impact

Mathematical models become an important and popular tools to understand the dynamics of the disease and give an insight to reduce the impact of malaria burden within the community. Thus, this paper aims to apply a mathematical model to study global stability of malaria transmission dynamics model with logistic growth. Analysis of the model applies scaling and sensitivity analysis and sensitivity analysis of the model applied to understand the important parameters in transmission and prevalence of malaria disease. We derive the equilibrium points of the model and investigated their stabilities. The results of our analysis have shown that if R0≤1, then the disease-free equilibrium is globally asymptotically stable, and the disease dies out; if R0>1, then the unique endemic equilibrium point is globally asymptotically stable and the disease persists within the population. Furthermore, numerical simulations in the application of the model showed the abrupt and periodic variations.

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Mathematical analysis of an age structured epidemic model with a quarantine class

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2021049 ◽

2021 ◽

Author(s):

Bedreddine AINSEBA ◽

Tarik Touaoula ◽

Zakia Sari

Keyword(s):

Reproduction Number ◽

Asymptotic Behavior Of Solutions ◽

Model Parameters ◽

Qualitative Behavior ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Age Structured ◽

The Stability ◽

The Impact

In this paper, an age structured epidemic Susceptible-Infected-Quarantined-Recovered-Infected (SIQRI) model is proposed, where we will focus on the role of individuals that leave their class of quarantine before being completely recovered and thus will participate again to the transmission of the disease. We investigate the asymptotic behavior of solutions by studying the stability of both trivial and positive equilibria. In order to see the impact of the different model parameters like the relapse rate on the qualitative behavior of our system, we firstly, give the explicit expression of the epidemic reproduction number $R_{0}.$ This number is a combination of the classical epidemic reproduction number for the SIQR model and a new epidemic reproduction number corresponding to the individuals infected by a relapsed person from the R-class. It is shown that, if $R_{0}\leq 1$, the disease free equilibrium is globally asymptotically stable and becomes unstable for $R_{0}>1$. Secondly, while $R_{0}>1$, a suitable Lyapunov functional is constructed to prove that the unique endemic equilibrium is globally asymptotically stable on some subset $\Omega_{0}.$

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Markov Chain Monte Carlo Analysis of the Variable-Volume Exothermic Model for a Continuously Stirred Tank Reactor

Engineering, Technology & Applied Science Research ◽

10.48084/etasr.3962 ◽

2021 ◽

Vol 11 (2) ◽

pp. 6919-6929

Author(s):

J. P. Muhirwa ◽

S. I. Mbalawata ◽

V. G. Masanja

Keyword(s):

Rank Correlation ◽

Least Square ◽

Stirred Tank ◽

Physical Parameters ◽

Model Parameters ◽

Stirred Tank Reactor ◽

Tank Reactor ◽

Variable Volume ◽

Continuously Stirred Tank Reactor ◽

The Impact

In this paper, a variable-volume Continuously Stirred Tank Reactor (CSTR) deterministic exothermic model has been formulated based on the Reynold Transport Theorem. The numerical analysis of the formulated model and the identifiability of its physical parameters are done by using the least squares and the Delayed-Rejection Adaptive Metropolis (DRAM) method. The least square estimates provide the prior information for the DRAM method. The overall numerical results show that the model gives an insight in describing the dynamics of CSTR processes, and 14 parameters of the CSTR are well identified through DRAM convergence diagnostic tests, such as trace, scatter, autocorrelation, histograms, and marginal density plots. Global sensitivity analysis was further performed, by using the partial rank correlation coefficients obtained from the Latin hypercube sampling method, in order to study and quantify the impact of estimated parameters, uncertainties on the model outputs. The results showed that 7 among the 14 estimated model parameters are very sensitive to the model outcomes and so those parameters need to be handled and treated carefully.

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Mathematical Modeling for Prediction Dynamics of the Coronavirus Disease 2019 (COVID-19) Pandemic, Quarantine Control Measures

Symmetry ◽

10.3390/sym12091404 ◽

2020 ◽

Vol 12 (9) ◽

pp. 1404

Author(s):

Din Prathumwan ◽

Kamonchat Trachoo ◽

Inthira Chaiya

Keyword(s):

Endemic Equilibrium ◽

Control Measures ◽

Single Infection ◽

Model Parameters ◽

Asymptotically Stable ◽

Expected Number ◽

Susceptible Population ◽

Proposed Model ◽

The Stability ◽

Disease Free

A mathematical model for forecasting the transmission of the COVID-19 outbreak is proposed to investigate the effects of quarantined and hospitalized individuals. We analyze the proposed model by considering the existence and the positivity of the solution. Then, the basic reproduction number (R0)—the expected number of secondary cases produced by a single infection in a completely susceptible population—is computed by using the next-generation matrix to carry out the stability of disease-free equilibrium and endemic equilibrium. The results show that the disease-free equilibrium is locally asymptotically stable if R0<1, and the endemic equilibrium is locally asymptotically stable if R0>1. Numerical simulations of the proposed model are illustrated. The sensitivity of the model parameters is considered in order to control the spread by intervention strategies. Numerical results confirm that the model is suitable for the outbreak that occurred in Thailand.

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Mathematical Analysis of Malaria-Schistosomiasis Coinfection Model

Epidemiology Research International ◽

10.1155/2016/3854902 ◽

2016 ◽

Vol 2016 ◽

pp. 1-19 ◽

Cited By ~ 2

Author(s):

E. A. Bakare ◽

C. R. Nwozo

Keyword(s):

Numerical Simulations ◽

Endemic Equilibrium ◽

Model Parameters ◽

Asymptotically Stable ◽

Reproduction Numbers ◽

Mathematical Techniques ◽

Parameter Values ◽

The Impact ◽

Number Of Individuals ◽

Disease Free

We formulated and analysed a mathematical model to explore the cointeraction between malaria and schistosomiasis. Qualitative and comprehensive mathematical techniques have been applied to analyse the model. The local stability of the disease-free and endemic equilibrium was analysed, respectively. However, the main theorem shows that if RMS<1, then the disease-free equilibrium is locally asymptotically stable and the phase will vanish out of the host and if RMS>1, a unique endemic equilibrium is also locally asymptotically stable and the disease persists at the endemic steady state. The impact of schistosomiasis and its treatment on malaria dynamics is also investigated. Numerical simulations using a set of reasonable parameter values show that the two epidemics coexist whenever their reproduction numbers exceed unity. Further, results of the full malaria-schistosomiasis model also suggest that an increase in the number of individuals infected with schistosomiasis in the presence of treatment results in a decrease in malaria cases. Sensitivity analysis was further carried out to investigate the influence of the model parameters on the transmission and spread of malaria-schistosomiasis coinfection. Numerical simulations were carried out to confirm our theoretical findings.

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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.

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Global Stability Analysis of Two-Stage Quarantine-Isolation Model with Holling Type II Incidence Function

Mathematics ◽

10.3390/math7040350 ◽

2019 ◽

Vol 7 (4) ◽

pp. 350 ◽

Cited By ~ 2

Author(s):

Mohammad A. Safi

Keyword(s):

Disease Transmission ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Control Measures ◽

Asymptotically Stable ◽

Two Stage ◽

Globally Asymptotically Stable ◽

Incidence Function ◽

Special Case ◽

Disease Free

A new two-stage model for assessing the effect of basic control measures, quarantine and isolation, on a general disease transmission dynamic in a population is designed and rigorously analyzed. The model uses the Holling II incidence function for the infection rate. First, the basic reproduction number ( R 0 ) is determined. The model has both locally and globally asymptotically stable disease-free equilibrium whenever R 0 < 1 . If R 0 > 1 , then the disease is shown to be uniformly persistent. The model has a unique endemic equilibrium when R 0 > 1 . A nonlinear Lyapunov function is used in conjunction with LaSalle Invariance Principle to show that the endemic equilibrium is globally asymptotically stable for a special case.

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