Stability Of Disease-free Equilibrium Research Articles

On the Optimal Control of HIV-TB Co-Infection and Improvement of Workplace Productivity

Human immunodeficiency virus (HIV) and tuberculosis (TB) have long been known to have a synergistic relationship. This is a result of each of the diseases impacting negatively on the immune system of the infected persons. The impact of these diseases on workforce productivity is studied in this paper from the viewpoint of dynamical systems. In this paper, we present a nonlinear ordinary differential equation model to study the dynamics of HIV-TB co-infection and its effect on workforce productivity. The main model is first decoupled into two basic submodels of HIV-only and TB-only models, whose qualitative properties are presented before the qualitative properties of the main model are studied. While the HIV-only model is shown to have a globally asymptotically stable disease-free equilibrium whenever its basic reproduction number is less than unity, the TB-only model is shown to exhibit backward bifurcation under some conditions. To investigate the impact of various intervention strategies on the control of the co-infection and improvement of workforce productivity, five time-dependent controls (involving transmission prevention for the two diseases, therapy for the two diseases, and capacity building for improved workforce productivity) are incorporated into the basic model to form an optimal control problem, which is qualitatively analyzed using Pontryagin’s maximum principle and numerically simulated. Incremental cost-effectiveness analysis is conducted with the results of the numerical simulations. It is observed that the most cost-effective strategy for fighting the spread of the co-infection with enhanced productivity is that of combining both preventative and curative measures along with skills training.

SENSITIVITY ANALYSIS AND IMPACT OF AN IMPERFECT VACCINE OF TWO STRAINS OF HEPATITIS B VIRUS INFECTION

A mathematical model considering two strains of hepatitis B virus (HBV) chronic carriers, to assess the impact of dose-structured imperfect vaccine, in a population, is designed and analyzed. The model is shown to have a locally and globally asymptotically stable disease-free equilibrium (DFE) whenever its associated reproduction number is numerically less than unity. Numerical analysis of the model shows that with the expected 50% minimum efficacy of the first vaccine dose, vaccinating 55% of the susceptible population with the first vaccine dose will be sufficient to effectively control the spread of hepatitis B infection. Such effective control can also be achieved if 50% of the first vaccine dose recipients take the second dose. Threshold analysis reveals that an imperfect HBV vaccine should have positive or negative population-level effect. Latin hypercube sampling–PRCC analysis illustrates that disease transmission rate, birth rate, natural death rate and proportion of children born with maternal immunity are most influential parameters in the disease dynamics. In this paper, the sensitivity analysis based on mathematical and in addition statistical techniques have been performed to determine the significance of the model parameters. It is observed that a number of the parameters play an important role to determine the magnitude of the basic reproduction number. Sensitivity analysis is achieved to determine model parameters’ importance in disease dynamics. It is observed that the reproduction number is the most responsive quantity to the potent transmission rate of HBV and in addition also vital to control the spread of the disease.

Modeling and analysis of tuberculosis and pneumonia co-infection dynamics with cost-effective strategies

Tuberculosis and Pneumonia are among the major airborne diseases that shares common clinical traits with high human morbidity and mortality impacts. Thus, the co-infection of the two diseases becomes unavoidable in a community of comorbid populations. Due to scare resources, the huge disease burden co-imposed by these endemics requires an effective intervention to cushion the effect. To achieve this goal, cost–benefit analysis for the optimal controls is deemed necessary. In this study, an SEIR model for the co-dynamics of tuberculosis and pneumonia is formulated. The model was shown to have a globally stable disease-free (endemic) equilibrium where the reproduction number is less (greater) than unity. Sensitivity analysis by partial rank correlation coefficients shows that increasing infection recovery is only one among the three most influential parameters that reduces the magnitude of the diseases. Through Pontryagin′s maximum principle, the number of infected individuals and cost of controls (public health campaign, chemoprophylaxis, oral fluid intake and treatment) was minimized. In implementing the cost–benefit analysis, we employ three techniques: the Infection Averted Ratio (IAR), the Average Cost-effectiveness Ratio (ACER) and the Incremental Cost-effectiveness Ratio (ICER). On comparing the strategies considering cases (A–E), the ICER results showed that, strategy with treating tuberculosis alone is the most effective and cost-saving strategy. But in terms of infections averted, strategy with all the control measures is the best; however, unaffordable since it is highly expensive. We hope that, the findings of this study should help policy-makers when dealing with the co-existence of tuberculosis and pneumonia.

MATHEMATICAL ANALYSIS OF TRANSMISSION DYNAMICS OF HIV/ AIDS IN RISK STRUCTURED POPULATION

A model for assessing the impact of key population at risk in spreading HIV/ AIDS is designed. The model is shown to have a globally asymptotically stable (GAS) disease-free equilibrium whenever the associated reproduction number is less than unity. It has a unique GAS endemic equilibrium whenever reproduction number exceeds unity, if there is no back and forth movement in susceptible classes. It is shown that if infected individuals with high risk abandon their risky behavioral practices, then the disease burden will reduce. Furthermore, if infected individuals with low risk do not adopt risky behaviors then the transmission of the disease will decrease. Partial Rank Correlation Coefficient (PRCC) indices are assessed using all the infection-related classes and basic reproduction number of the model. Uncertainty and sensitivity analysis results show that controlling the effective contact rate, disease-induced death rate in AIDS stage with low and high risk classes will help controlling the transmission of the disease.

Dynamic analysis and optimal control of worm propagation model with saturated incidence rate

In order to prevent the propagation of computer worms effectively, based on the latent character of worms, the exposed compartments of computer and USB device are introduced respectively, and a series of computer worm propagation models with saturation incidence rate are proposed. The qualitative behavior of the proposed model is studied. Firstly, the threshold R0 of the model is derived by using the next-generation matrix method, which completely characterized the stability of disease free equilibrium and endemic equilibrium. If R0 < 1, the disease free equilibrium is asymptotically stable, implying that the worm dies out eventually and its attack remains under control; if R0 > 1, the asymptotic stability of endemic equilibrium under certain conditions is proved, which means that the worm is always persistent and uncontrollable under such conditions. Secondly, the theoretical results are verified by numerical study, in which the relative importance of each parameter in worm prevalence is evaluated by sensitivity analysis. Finally, so as to minimize the number of computer and USB device carrying computer worms in short span of time, the worm propagation model is extended to incorporate three control strategies. The Pontryagin’s maximum principle is used to characterize the controls’ optimal levels. According to the control effect diagram, the combined strategy is effective in minimizing the transmission dynamics of worm virus in computer and USB devices populations respectively.

Global Stability of Covid-19 Disease Free Based on Sivrs Model

This study discusses the spread of the Covid-19 disease by including new variant variables. The model used by SIVRS assumes there are deaths caused by Covid-19 and the new variant Covid-19. In addition, individuals who have been infected with the new variant of Covid-19 can recover. Based on the model, disease-free equilibrium points and endemic equilibrium points are obtained. The analysis was carried out around the disease-free equilibrium point and the result was that the global asymptotically stable disease-free equilibrium point with the condition R0<1. Furthermore, a simulation was carried out with the Maple 18.

Modelling the effect of environmental pollution on Zika outbreak: A case study of Brazil

In this paper, we formulate a compartmental model to study the Zika Epidemic in Brazil. The goal of this work is to understand if pollution plays any role in the spread of Zika Epidemic. Some theories are there in the literature that show that there might be a direct relation between Zika spread and environment contamination. However, it is very hard to find real-world data to justify these claims. In the present study, we have used the official data and have provided a direct link to fill that void. For the study, we have considered the combination of the human population and mosquito population. We have provided a complete theoretical study of the model. For the proposed model we have derived the basic reproduction number, stability analysis as well as bifurcation analysis corresponding to the basic reproduction number $ R_0 $. The stability of disease free equilibrium corresponding to $ R_0 < 1 $ implies that the disease can be eradicated provided $ R_0 $ does not exceed 1. We have also implemented PRCC technique to perform global uncertainty analysis. Our numerical simulation shows the impact of various sensitive parameters on $ R_0 $. We have also shown how the invectives will amplify different scenarios. Also, we have gathered real data and shown a direct relation of environmental contamination of the disease spread.

Optimal Control and Bifurcation Analysis of HIV Model.

In this study, a very crucial stage of HIV extinction and invisibility stages are considered and a modified mathematical model is developed to describe the dynamics of infection. Moreover, the basic reproduction number R 0 is computed using the next-generation matrix method whereas the stability of disease-free equilibrium is investigated using the eigenvalue matrix stability theory. Furthermore, if R 0 ≤ 1, the disease-free equilibrium is stable both locally and globally whereas if R 0 > 1, based on the forward bifurcation behavior, the endemic equilibrium is locally and globally asymptotically stable. Particularly, at the critical point R 0 = 1, the model exhibits forward bifurcation behavior. On the other hand, the optimal control problem is constructed and Pontryagin's maximum principle is applied to form an optimality system. Further, forward fourth-order Runge-Kutta's method is applied to obtain the solution of state variables whereas Runge-Kutta's fourth-order backward sweep method is applied to obtain solution of adjoint variables. Finally, three control strategies are considered and a cost-effective analysis is performed to identify the better strategies for HIV transmission and progression. In advance, prevention control measure is identified to be the better strategy over treatment control if applied earlier and effectively. Additionally, MATLAB simulations were performed to describe the population's dynamic behavior.

Controlling malaria in a population accessing counterfeit antimalarial drugs.

A mathematical model is developed for describing malaria transmission in a population consisting of infants and adults and in which there are users of counterfeit antimalarial drugs. Three distinct control mechanisms, namely, effective malarial drugs for treatment and insecticide-treated bednets (ITNs) and indoor residual spraying (IRS) for prevention, are incorporated in the model which is then analyzed to gain an understanding of the disease dynamics in the population and to identify the optimal control strategy. We show that the basic reproduction number, $ R_{0} $, is a decreasing function of all three controls and that a locally asymptotically stable disease-free equilibrium exists when $ R_{0} < 1 $. Moreover, under certain circumstances, the model exhibits backward bifurcation. The results we establish support a multi-control strategy in which either a combination of ITNs, IRS and highly effective drugs or a combination of IRS and highly effective drugs is used as the optimal strategy for controlling and eliminating malaria. In addition, our analysis indicates that the control strategies primarily benefit the infant population and further reveals that a high use of counterfeit drugs and low recrudescence can compromise the optimal strategy.

Stability analysis and optimal control of worm propagation model with saturated incidence rate

Worms spreading through Web-based scanning and removable devices pose a serious threat to Internet security. This paper is devoted to solving the problems of computer worms. A nonlinear mathematical model of the problem is established, and the characteristics and mechanisms of worm propagation are analyzed by means of the stability theory of differential equations, optimal control and computer simulation. The results provide some new insights to computer security, which is to predict the tendency of worm propagation through the stability of endemic equilibrium, identify the epidemic control strategies by the stability of disease-free equilibrium, and evaluate the prevalence of worms based on the final scale of infected devices. The performance of our model is evaluated by numerical simulation, and the results indicate that our combined strategy can combat the worm propagation. The results may be helpful to inhibit the worm spread widely in the network.

Appraisal and Simulation on Codynamics of Pneumonia and Meningitis with Vaccination Intervention: From a Mathematical Model Perspective.

The membranes that encompass the brain and spinal cord become inflamed by the potentially fatal infectious disease called pneumococcal meningitis. Pneumonia and meningitis "coinfection" refers to the presence of both conditions in a single host. In this work, we accounted for the dynamics of pneumonia and meningitis coinfection in communities by erroneously using a compartment model to analyze and suggest management techniques to stakeholders. We have used the next generation matrix approach and derived the effective reproduction numbers. When the reproduction number is less than one, the constructed model yields a locally asymptotically stable disease-free equilibrium point. Additionally, we conducted a sensitivity analysis to determine how different factors affected the incidence and transmission rate, which revealed that both the pneumonia and meningitis transmission rates are extremely sensitive. The performance of our numerical simulation demonstrates that the endemic equilibrium point of the pneumonia and meningitis coinfection model is locally asymptotically stable when max{ ℛ 1, ℛ 2} > 1. Finally, as preventative and control measures for the coinfection of pneumonia and meningitis illness, the stakeholders must concentrate on reducing the transmission rates, reducing vaccination wane rates, and boosting the portion of vaccination rates for both pneumonia and meningitis.

Effect of awareness, quarantine and vaccination as control strategies on COVID-19 with Co-morbidity and Re-infection.

In this paper, a deterministic compartmental model is presented to assess the impact of vaccination and non-pharmaceutical interventions (social distance, awareness, face mask, and quarantine) on the transmission dynamics of COVID-19 with co-morbidity and re-infection. An expression for the basic reproduction number is then derived for this model. Theoretical analysis shows that the model exhibits backward bifurcation phenomenon when the basic reproduction number is less than unity. But for the case of no re-infection, the model has a globally asymptotically stable disease-free equilibrium (DFE) when the basic reproduction number is less than unity. Furthermore, it is shown that in the case of no re-infection, a unique endemic equilibrium point (EEP) of the model exists which is globally asymptotically stable whenever the reproduction number is greater than unity. From the global sensitivity and uncertainty analysis, we have identified mask coverage, mask efficacy, vaccine coverage, vaccine efficacy, and contact rate as the most influential parameters influencing the spread of COVID-19. Numerical simulation results show that the use of effective vaccines with proper implementation of non-pharmaceutical interventions could lead to the elimination of COVID-19 from the community. Numerical simulations also suggest that the control strategy that ensures a continuous and effective mass vaccination program is the most cost-effective control strategy. The study also shows that in the presence of any co-morbidity and with the occurrence of re-infection, the disease burden may increase.

Effect of saturated treatment on malaria spread with optimal intervention

This work presents a new deterministic model which considers transmission mode through blood transfusion and saturated treatment function with a view to gaining more insights into the dynamical spread of malaria. Existence of steady states of the model is established analytically. The key epidemiological threshold (R0) which determines the potential spread of malaria is computed using the next generation matrix method. The model has a locally asymptotically stable disease-free equilibrium point at R0<1, and condition which ensures the appearance of two endemic equilibria at R0<1 suggesting the possibility of backward bifurcation is determined. It is shown, using Lyapunov function, that a unique endemic equilibrium point is globally asymptotically stable at R0>1. Moreover, the model is modified by incorporating four time-dependent optimal control variables, namely insecticide-treated nets targeted at vector-human transmission route, screening of blood against human–human blood transfusion, anti-malaria drug treatment and vector control with insecticide spraying. Optimal control theory with the aid of Pontryagin’s maximum principle is applied to characterize the optimal control quadruple. Based on efficiency and cost-effectiveness analyses carried out, the most efficient and the most cost-effective combination of the four controls that can be implemented to reduce the burden of malaria in the population are obtained.

Study and Analysis of Corona-Virus Transfer Dynamics using Enhanced SEIR Epidemic Model with Vaccination Effect

Corona Virus Syndrome (COVID-19) is a contagious disease and it is considered one of the deadliest viruses ever known to humanity. In this work, the transmission dynamics of the COVID- 19 has been studied using an enhanced SEIR epidemic compartmental model with a vaccination compartment. This model divides the whole population into five categories: susceptible (S), exposed (E), infectious (I), recovered (R), and vaccinated (V). Firstly, Positivity, Existence and Uniqueness of solution are illustrated. Secondly, a mathematical analysis is done to study the equilibrium points of the model and the basic reproduction number has been computed using Next Generation Matrix method. Thirdly, by using analytical computation the Stability of disease free equilibrium (DFE) point of the model is discussed and the Existence, and uniqueness of the endemic equilibrium are proved. Finally, Bifurcation coefficient is computed using the available data and Simulations are made to illustrate the effect of the transmission rate and vaccination rate on the value of the basic reproduction number.

Post pandemic fatigue: what are effective strategies?

Recurrent updates in non-pharmaceutical interventions (NPIs) aim to control successive waves of the coronavirus disease 2019 (COVID-19) but are often met with low adherence by the public. This study evaluated the effectiveness of gathering restrictions and quarantine policies based on a modified Susceptible-Exposed-Infectious-Hospitalized-Recovered (SEIHR) model by incorporating cross-boundary travellers with or without quarantine to study the transmission dynamics of COVID-19 with data spanning a nine-month period during 2020 in Hong Kong. The asymptotic stability of equilibria reveals that the model exhibits the phenomenon of backward bifurcation, which in this study is a co-existence between a stable disease-free equilibrium (DFE) and an endemic equilibrium (EE). Even if the basic reproduction number ({mathscr {R}}_{0}) is less than unity, this disease cannot be eliminated. The effect of each parameter on the overall dynamics was assessed using Partial Rank Correlation Coefficients (PRCCs). Transmission rates (i.e., beta _1 and beta _2), effective contact ratio a_6 between symptomatic individuals and quarantined people, and transfer rate theta _3 related to infection during quarantine were identified to be the most sensitive parameters. The effective contact ratios between the infectors and susceptible individuals in late July were found to be over twice as high as that in March of 2020, reflecting pandemic fatigue and the potential existence of infection during quarantine.

Cost Effective Analysis on Mathematical Modelling of HIV/AIDS with Optimal Control Strategy

In this paper, a deterministic model of the Human Immunodeficiency Virus has been formulated to describe the transmission dynamics of the disease. The good posedness of the model equations was proved and the equilibrium points of the model have been identified. Basic reproduction numbers were used to establish both local and global stability of the disease-free and endemic equilibrium points of the model equation. The analysis reveals that if the basic reproduction is smaller than one, the solution converges to the disease-free steady-state, which is locally asymptotically stable. If the fundamental reproduction number is more than one, the solution converges to the endemic equilibrium point, which is locally asymptotically stable., sensitivity analysis of the model equation was performed on the key parameters to find out their relative significance and potential impact on the transmission dynamics of the Human Immunodeficiency Virus. The results of the simulation show that treatment minimizes the risk of Human Immunodeficiency Virus transmission from the community and the stability of disease-free equilibrium is achievable when basic reproduction is less. The findings from the analysis of cost-effectiveness revealed that a combination of prevention and screening is the most effective strategy to eradicate the disease from the community.

Analysis of SEIR epidemic patch model with nonlinear incidence rate, vaccination and quarantine strategies

In this paper, a susceptible–exposed–infectious–recovered (SEIR) epidemic patch model with nonlinear incidence rate, vaccination and quarantine strategies is presented. According to the properties of nonnegative matrices, the range of basic reproduction number is obtained. Moreover, the stability of disease-free equilibrium is studied. Some numerical simulations are given to validate the theoretical results. The effects of vaccination, quarantine strategies and population migration on disease transmission are also investigated by numerical simulations.

Mathematical analysis and control of chikungunya (CHIKV) virus by SEIR model

Chikungunya is a re-emerging arboviral disease in Asia and Africa infected by Aedes mosquitoes which had posed a global threat in several countries. Vector borne diseases are the primary cause of death in most of the world countries, hence it becomes pertinent to control these vector borne diseases. In this paper a mathematical model is provided by dividing it in four components namely Susceptible human, Exposed human, Infected human, Recovered human.The study is carried out on basic reproduction number and stability analysis. A mathematical model is developed for globally asymptotically stable disease-free equilibrium, when the associated reproduction number is less than unity. The aim of this study is to formulate a model where number of infectives will not change and the infection rate equals to the recovery rate by reaching stable endemic equilibrium point.

Towards the global equilibrium of COVID-19: Statistical analysis of country-level data.

In our study, we explore the COVID‐19 dynamics to test whether the virus has reached its equilibrium point and to identify the main factors explaining the current R and case fatality rate (CFR) variability across countries. We present a retrospective study of publicly available country‐level data from 50 countries having the highest number of confirmed COVID–19 cases at the end of September 2021. The mean values of country‐level moving averages of R and CFR went down respectively from 1.118 and 6.3% on June 30, 2020 to 1.083 and 3.6% on September 30, 2020 and to 1.015 and 1.8% by September 30, 2021. In parallel, the 10%–90% inter‐percentile range of R and CFR moving averages decreased, respectively, from 0.288 and 13.3% on June 30, 2020, to 0.151 and 7.7% on September 30, 2020, and to 0.107 and 3.3% by September 30, 2021. The slow decrease in the country‐level moving averages of R, approaching the level of 1.0 and accompanied by repeated outbreaks (“waves”) in various countries, may indicate that COVID‐19 has reached its point of stable endemic equilibrium. A regression analysis implies that only a prohibitively high level of herd immunity (about 63%) may stop the endemic by reaching a stable disease‐free equilibrium. It also appears that fully vaccinating about 70% of a country's population should be sufficient for bringing the CFR close to the level of the seasonal flu (about 0.1%). Thus, while the currently available vaccines prove to be effective in reducing the mortality from the existing COVID‐19 variants, they are unlikely to stop the spread of the virus in the foreseeable future. It is noteworthy that government measures restricting people's behavior (such as lockdowns) were not found to have statistically significant effects in the analyzed data.

A study on the maize streak virus epidemic model by using optimized linearization-based predictor-corrector method in Caputo sense

In this article, we study the dynamics of a Maize streak virus (MSV) epidemic model by using the Caputo fractional derivative. Firstly, we define the dynamics of the given fractional-order model by checking the non-negativity and boundedness of the solution, stability of disease-free equilibrium, the existence of a unique solution, and its stability. Then we derive the numerical solution of the proposed model by using an optimized Predictor-Corrector method, which has not yet been applied to solve any kind of epidemic system until now. Our optimized method uses a linear approximation of the proposed nonlinear model to ameliorate the competence of the Predictor-Corrector schemes. To verify the correctness of our results, we plot various graphs by taking different parameter cases at various fractional-order values. Also, Caputo outputs are compared with Atangana-Baleanu-Caputo derivative outputs. Our research shows the usefulness of the proposed optimized Predictor-Corrector method in epidemic studies and for simulating the memory effects in the given MSV system. The solution methodology of the proposed system is the main novelty of this research along with other supporting analyses.