Stability Of Disease-free Equilibrium Research Articles

Significance of antiviral therapy and CTL-mediated immune response in containing hepatitis B and C virus infection

Viral infections remain a major cause of deaths globally. Here, we focus on Hepatitis B and C viruses (HBV and HCV) infection dynamics in the liver and blood cells, taking into account non-cytotoxic and cytotoxic mediated immune response as well as antiviral therapy. The analysis of the model is presented in terms of the reproduction number, R0. The system has a globally stable disease-free equilibrium when R0 is below unit. It exhibits forward bifurcation when R0 is greater than one, with a locally asymptotically stable endemic equilibrium point. By carrying out sensitivity analysis using the Latin Hypercube Sampling scheme (LHS), we determine the key processes that are essential in containing the infection. Our results suggest that, the production of virions from blood and liver cells as well as degeneration of Cytotoxic T-Lymphocytes (CTLs) can greatly aggravate HBV and HCV infections. Moreover, CTL-mediated immune response alone cannot effectively contain the infection. Although virion production from the liver and blood as well as per-capita loss of CTLs were observed to greatly aggravate the infection, their effects imposed on the infection dynamics were not significantly different. We recommend based on the results that, use of highly effective antiviral/combination therapy to block production of new virions is an ideal approach to containing the infections.

Theoretical perspectives on the within-human host model for Lassa fever

In this paper, we theoretically analyzed the within-human host model for Lassa fever earlier proposed by Obasi and Mbah, so as to understand the dynamics of Lassa fever transmission at the human population level. The model has locally asymptotically stable disease-free equilibrium whenever the associated reproduction number is less than unity. This model will undergo the phenomenon of backward bifurcation where the stable disease-free co-exists with a stable endemic equilibrium, when the associated reproduction number is less than unity. This implies that bringing down the reproduction number to below unity is not enough to eradicate Lassa fever disease within human population. It is also shown that the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction number is less than unity. The reproduction number, RWH<1, which is an important parameter in the control of Lassa fever infection, has been calculated using the next generation method. We have also shown that the endemic equilibrium point exists for RWN>1 and has been noted that this endemic equilibrium is unique and locally asymptotically stable based on Lyaponuv Function. However, this work has thrown up important parameters that could be gathered by the relevant government agencies for better understanding of the burden of Lassa fever disease in the human population.

Epidemic Model Analysis of Covid-19

Covid-19 is a very extraordinary case not only in one country but all countries in the world. The number of deaths caused by Covid-19 is very large and the rate of spread of this disease is very high and fast. In this paper, we perform an analysis of a covid-19 epidemic model. This model is a development of the SEIR model in general which is equipped with a Quarantine (Q), Fatality (F) compartment, and there is a separation between detected and undetected infected people (I). Our analysis shows that there are two equilibria, namely, disease free equilibrium and endemic equilibrium. by using, Lyapunov function, we demonstrated that disease free is globally asymptotically stable if R0 &lt; 1, and disease-free becomes unstable if R0 &gt; 1. This result reveal that the intervention of infection rate and quarantine process are important to control and achieve global stability of disease-free equilibrium

Model matematika SMEIUR pada penyebaran penyakit campak dengan faktor pengobatan

This study discusses the spread of measles in a mathematical model. Mathematical modeling is not only limited to the world of mathematics but can also be applied in the health sector. Measles is a disease with a high transmission rate. The spread of measles in this model was modified by adding the treated population and the treatment parameters of the exposed population. In this article, we examine the equilibrium points in the SMEIUR mathematical model and perform stability analysis and numerical simulations. In this study, two equilibrium points were obtained, namely the disease-free and endemic equilibrium point. After getting the equilibrium point, an analysis is carried out to find the stability of the model. Furthermore, the simulation produces a stable disease-free equilibrium point at conditions R01 and a stable endemic equilibrium point at conditions R01. In this study, a numerical simulation was carried out to see population dynamics by varying the parameter values. The simulation results show that to reduce the spread of measles, it is necessary to increase the rate of advanced immunization, the rate of the infected population undergoing treatment, and the proportion of individuals who are treated cured.

Stability analysis and optimal control of covid-19 with convex incidence rate in Khyber Pakhtunkhawa (Pakistan)

The dynamic of covid-19 epidemic model with a convex incidence rate is studied in this article. First, we formulate the model without control and study all the basic properties and results including local and global stability. We show the global stability of disease free equilibrium using the method of Lyapunov function theory while for disease endemic, we use the method of geometrical approach. Furthermore, we develop a model with suitable optimal control strategies. Our aim is to minimize the infection in the host population. In order to do this, we use two control variables. Moreover, sensitivity analysis complemented by simulations are performed to determine how changes in parameters affect the dynamical behavior of the system. Taking into account the central manifold theory the bifurcation analysis is also incorporated. The numerical simulations are performed in order to show the feasibility of the control strategy and effectiveness of the theoretical results.

A DELAYED DISCRETE MULTI-GROUP NONLINEAR EPIDEMIC MODEL WITH VACCINATION AND LATENCY

A delayed discrete multi-group nonlinear epidemic model with vaccination and latency is derived by the application of a nonstandard finite difference scheme. It is proved that the extinction and persistence of the disease are determined by a threshold parameter in term of the basic reproduction number $ R $. More precisely, when $ R\leq 1 $ the disease goes to extinction with the globally asymptotically stable disease-free equilibrium, while the disease is persistent with the globally asymptotically stable endemic equilibrium when $ R&gt;1 $.

Optimal Strategies for Control of COVID-19: A Mathematical Perspective.

A deterministic ordinary differential equation model for SARS-CoV-2 is developed and analysed, taking into account the role of exposed, mildly symptomatic, and severely symptomatic persons in the spread of the disease. It is shown that in the absence of infective immigrants, the model has a locally asymptotically stable disease-free equilibrium whenever the basic reproduction number is below unity. In the absence of immigration of infective persons, the disease can be eradicated whenever ℛ0 < 1. Specifically, if the controls ui, i=1,2,3,4, are implemented to 100% efficiency, the disease dies away easily. It is shown that border closure (or at least screening) is indispensable in the fight against the spread of SARS-CoV-2. Simulation of optimal control of the model suggests that the most cost-effective strategy to combat SARS-CoV-2 is to reduce contact through use of nose masks and physical distancing.

Mathematical modeling of COVID-19: Impact of non-pharmaceutical interventions in India.

In the absence of effective vaccine/antiviral strategies for reducing the burden of the coronavirus disease 2019 (COVID-19) pandemic in India, the main focus has been on basic non-pharmaceutical interventions (NPIs), such as nationwide lockdown (travel restrictions and the closure of schools, shopping malls, and worshipping and other gathering places), quarantining of exposed individuals, and isolation of infected individuals. In the present study, we propose a compartmental epidemic model incorporating quarantine and isolation compartments to (i) describe the current transmission patterns of COVID-19 in India, (ii) assess the impact of currently implemented NPIs, and (iii) predict the future course of the pandemic with various scenarios of NPIs in India. For R0<1, the system has a globally asymptotically stable disease free equilibrium, while for R0>1, the system has one unstable disease free equilibrium and a unique locally stable endemic equilibrium. By using the method of least squares and the best fit curve, we estimate the model parameters to calibrate the model with daily new confirmed cases and cumulative confirmed cases in India for the period from May 1, 2020 to June 25, 2020. Our result shows that the implementation of an almost perfect isolation in India and 33.33% increment in contact-tracing on June 26, 2020 may reduce the number of cumulative confirmed cases of COVID-19 in India by around 53.8% at the end of July 2020. Nationwide lockdown with high efficiency can diminish COVID-19 cases drastically, but combined NPIs may accomplish the strongest and most rapid impact on the spreading of COVID-19 in India.

Mathematical modelling and optimal cost-effective control of COVID-19 transmission dynamics.

The novel coronavirus disease (COVID-19) caused by a new strain of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) remains the current global health challenge. In this paper, an epidemic model based on system of ordinary differential equations is formulated by taking into account the transmission routes from symptomatic, asymptomatic and hospitalized individuals. The model is fitted to the corresponding cumulative number of hospitalized individuals (active cases) reported by the Nigeria Centre for Disease Control (NCDC), and parameterized using the least squares method. The basic reproduction number which measures the potential spread of COVID-19 in the population is computed using the next generation operator method. Further, Lyapunov function is constructed to investigate the stability of the model around a disease-free equilibrium point. It is shown that the model has a globally asymptotically stable disease-free equilibrium if the basic reproduction number of the novel coronavirus transmission is less than one. Sensitivities of the model to changes in parameters are explored, and safe regions at certain threshold values of the parameters are derived. It is revealed further that the basic reproduction number can be brought to a value less than one in Nigeria, if the current effective transmission rate of the disease can be reduced by 50%. Otherwise, the number of active cases may get up to 2.5% of the total estimated population. In addition, two time-dependent control variables, namely preventive and management measures, are considered to mitigate the damaging effects of the disease using Pontryagin’s maximum principle. The most cost-effective control measure is determined through cost-effectiveness analysis. Numerical simulations of the overall system are implemented in hbox {MatLab}^circledR for demonstration of the theoretical results.

A mathematical analysis of a tuberculosis epidemic model with two treatments and exogenous re-infection

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.

Mathematical analysis of the effects of controls on transmission dynamics of SARS-CoV-2

COVID-19, an infectious disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), starting from Wuhan city of China, plagued the world in the later part of 2019. We developed a deterministic model to study the transmission dynamics of the disease with two categories of the Susceptibles (ie Immigrant Susceptibles and Local Susceptible). The model is shown to have a globally stable disease-free equilibrium point whenever the basic reproduction number R0 is less than unity. The endemic equilibrium is also shown to be globally stable for R0>1 under some conditions. The spread of the disease is also shown to be highly sensitive to use of PPEs and personal hygiene (d), transmission probability (β), average number of contacts of infected person per unit time (day) (c), the rate at which the exposed develop clinical symptoms (δ) and the rate of recovery (ρ). Numerical simulation of the model is also done to illustrate the analytical results established.

The development of a deterministic dengue epidemic model with the influence of temperature: A case study in Malaysia

In this study, a deterministic mathematical model of the dengue transmission is developed by considering the effect of temperature on the transmission dynamics. It has a locally asymptotically stable disease-free equilibrium (DFE) point whenever the basic reproduction number (R0) is less than unity. This model has shown a possibility of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium point when R0 < 1. Using the entomological data of the Aedes mosquito population and the experimental data of the dengue transmission in Malaysia, R0 is evaluated at different temperatures, given that at 32∘C, R0 attain its maximum value. The solutions of the model show an oscillatory behaviour. Although the oscillations are unobservable in the state variable, they are present in the numerical solution. Finally, a fractional analogue of the dengue model is presented, and numerical solutions are performed to make a comparison with the integer-order model. A comparison of the two results reveals that the fractional-order model provides stable solutions as the oscillatory behaviour can be dampened. This paper provides numerical evidence that the dengue virus can spread in temperate areas where it is no longer limited to tropical and subtropical regions, and suggests that the fractional-order approach can be a great alternative in modelling dengue transmission dynamics.

Modeling the Impact of Vaccination, Screening, Treatment on the Dynamics of Pneumonia

In this paper we have presented a deterministic model for pneumonia transmission and we have used the model to avail the potential impact of therapy. The model is based on the vaccinated-susceptible-carrier-infected-recovered-susceptible compartmental structure and their possible interventions with the possibility of infected individual recovery from natural immunity. Here, we have modeled Pneumonia considering vaccination, screening and treatment with a system of nonlinear ordinary differential equation. The model reproduction number R0 is derived and the stability of the equilibria are derived. The stability of equilibrium points is analyzed. The results shows that there exists a locally stable disease free equilibrium points, E0 when R0&lt;1 and a unique endemic equilibrium E1, when R0&gt;1. Infection free point was found to be locally stable and if reproduction number is greater than unity, then there is unique endemic equilibrium point and if it is less than unity, the endemic equilibrium point is globally asymptotically stable and pneumonia will be eliminated.

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

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&lt;1, and the endemic equilibrium is locally asymptotically stable if R0&gt;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.

Occurrence of backward bifurcation and prediction of disease transmission with imperfect lockdown: A case study on COVID-19.

The outbreak of COVID-19 caused by SARS-CoV-2 is spreading rapidly around the world, which is causing a major public health concerns. The outbreaks started in India on March 2, 2020. As of April 30, 2020, 34,864 confirmed cases and 1154 deaths are reported in India and more than 30,90,445 confirmed cases and 2,17,769 deaths are reported worldwide. Mathematical models may help to explore the transmission dynamics, prediction and control of COVID-19 in the absence of an appropriate medication or vaccine. In this study, we consider a mathematical model on COVID-19 transmission with the imperfect lockdown effect. The basic reproduction number, R 0, is calculated using the next generation matrix method. The system has a disease-free equilibrium (DFE) which is locally asymptotically stable whenever R 0 < 1. Moreover, the model exhibits the backward bifurcation phenomenon, where the stable DFE coexists with a stable endemic equilibrium when R 0 < 1. The epidemiological implications of this phenomenon is that the classical epidemiological requirement of making R 0 less than unity is only a necessary, but not sufficient for effectively controlling the spread of COVID-19 outbreak. It is observed that the system undergoes backward bifurcation which is a new observation for COVID-19 disease transmission model. We also noticed that under the perfect lockdown scenario, there is no possibility of having backward bifurcation. Using Lyapunov function theory and LaSalle Invariance Principle, the DFE is shown globally asymptotically stable for perfect lockdown model. We have calibrated our proposed model parameters to fit daily data from India, Mexico, South Africa and Argentina. We have provided a short-term prediction for India, Mexico, South Africa and Argentina of future cases of COVID-19. We calculate the basic reproduction number from the estimated parameters. We further assess the impact of lockdown during the outbreak. Furthermore, we find that effective lockdown is very necessary to reduce the burden of diseases.

Modeling and Analysis of Effect of Awareness Programs by Media on the Spread of COVID-19 Pandemic Disease

This paper proposes and analyses a basic deterministic mathematical model to investigate Modeling and Analysis of effect of awareness program by media on the spread COVID-19 Pandemic Disease. The model has four non-linear differential equations, which describe the effects of awareness programs on the spread of COVID-19 Pandemic diseases such as flu has been proposed and analyzed. In the modeling process assumed that disease spreads due to the contact between susceptible and infective only. The growth rate of awareness programs influencing the population assumed to be proportional to the number of infective individuals. It further, assumed that due to the effect of media, susceptible individuals form a separate class and avoid contact with the infective. The model analyzed by using stability theory of differential equations. The model analysis shows that the spread of a COVID-19 Pandemic disease controlled by using awareness programs but the disease remains Pandemic due to immigration. The models analyzed qualitatively to determine criteria for control of the spread of COVID 19, and used to calculate the basic reproduction R0. The equilibrium of COVID 19 models is determined. In addition to having a disease-free equilibrium, which is globally asymptotically stable when the R0 < 1, the basic COVID 19 model manifest one's possession of (a quality of) the phenomenon of backward bifurcation where a stable disease-free equilibrium co-exists (at the same time) with a stable endemic equilibrium for a certain range of associated reproduction number less than one. The analysis and simulation results of the model suggested that the most effective strategies for controlling or eradicating the spread of COVID 19 pandemic were suggest using that awareness programs through the media campaigning are helpful in decreasing the spread of COVID 19 Pandemic diseases by isolating a fraction of susceptible from infective.

Analysis of a mathematical model for the transmission dynamics of human melioidosis

A deterministic model for the transmission dynamics of melioidosis disease in human population is designed and analyzed. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the basic reproduction number [Formula: see text] is less than one. It is further shown that the backward bifurcation dynamics is caused by the reinfection of individuals who recovered from the disease and relapse. The existence of backward bifurcation implies that bringing down [Formula: see text] to less than unity is not enough for disease eradication. In the absence of backward bifurcation, the global asymptotic stability of the disease-free equilibrium is shown whenever [Formula: see text]. For [Formula: see text], the existence of at least one locally asymptotically stable endemic equilibrium is shown. Sensitivity analysis of the model, using the parameters relevant to the transmission dynamics of the melioidosis disease, is discussed. Numerical experiments are presented to support the theoretical analysis of the model. In the numerical experimentations, it has been observed that screening and treating individuals in the exposed class has a significant impact on the disease dynamics.

Global dynamics of a model of hepatitis B virus infection in a sub-Saharan African rural area

We formulate and systematically study a deterministic compartmental model of Hepatitis B. This model has some important and novel features compared with the well-known basic model in the literature. Specifically, it takes into account the differential susceptibility that follows the vaccine formulation employing three-doses schedule. It points up the HbeAg status of carriers, their levels of viral replication, the fact that treatment being not curative is recommended only to a small proportion of chronic carriers, and finally the fact that only inactive carriers are able to recover from disease. The model has simple dynamical behavior which has a globally asymptotically stable disease-free equilibrium when the basic reproduction number [Formula: see text] and an endemic equilibrium when [Formula: see text]. By the use of Lyapunov functions, when it exists, we prove the global asymptotic stability of the endemic equilibrium under some conditions. Using data from Tokombere, a rural area in Cameroon, numerical simulations are performed. These numerical simulations first confirm analytical results, second they suggest that a policy based on treatment could not significantly impact the course of the infection. Third, they show as it is well known that vaccination is a very effective measure to control the infection. Furthermore, they show that neonatal vaccination influences more the course of infection than mass vaccination strategy. Nevertheless, they picture how much loss between consecutive doses of vaccine could be harmful. Finally, it is suggested that for a Sub-saharan African rural area, two-thirds of expected incidence of Hepatitis B virus infection and one third of expected prevalence of chronic carriers could be averted by 2030 if the birth dose vaccination becomes systematic and if mass vaccination rate increases to up 10%.

Asymptotic behavior and threshold of a stochastic SIQS epidemic model with vertical transmission and Beddington\u2013DeAngelis incidence

This paper investigates a deterministic and stochastic SIQS epidemic model with vertical transmission and Beddington–DeAngelis incidence. Firstly, for the corresponding deterministic system, the global asymptotic stability of disease-free equilibrium and the endemic equilibrium is proved through the stability theory. Secondly, for the stochastic system, the threshold conditions which decide the extinction or permanence of the disease are derived. By constructing suitable Lyapunov functions, we investigate the oscillation behavior of the stochastic system solution near the endemic equilibrium. The results of this paper show that there exists a great difference between the deterministic and stochastic systems, which implies that the large stochastic noise contributes to inhibiting the spread of disease. Finally, in order to validate the theoretical results, a series of numerical simulations are presented.

Impact of Combined Therapy in HIV-1 Treatment: A Double Impulsive Approach

The major class of antiretroviral drugs used in the treatment of HIV positive patients are fusion inhibitors (FIs) and protease inhibitors (PIs). The fusion inhibitor enfuvirtide is effective in early viral life cycle and prevents viral entry into host cells. Protease inhibitors efficiently reduce the number of infectious virus particles. In this article, a mathematical model has been derived to study the effect of enfuvirtide in combination with protease inhibitors in HIV treatment. The dynamics of the two drugs have been analysed with a double impulsive control approach. We have considered different impulse interval for each drug and determine the threshold value for the dosages and dosing intervals to ensure the stability of disease free equilibrium. This study suggests that FIs-and-PIs combination therapy can have a better outcome than single drug activity; furthermore it can also be a better treatment strategy than the current combination of drugs when applied with proper dose and dosing intervals.