Stability Of Disease-free Equilibrium Point Research Articles

Nonlinear dynamics and optimal control strategies of a novel fractional-order lumpy skin disease model

Fractional order differential equations are often employed to better and accurately model several natural and physical processes with memory characteristics than the integer-order models. In this work, a proposed fractional-order model is presented to simulate one of the most dangerous viral cattle diseases. The present fractional-order Lumpy Skin Disease model considers the interactions between different compartments of cattle and virus vectors. The solution of the model is examined in terms of existence, uniqueness, positivity and boundedness. Then, the stability analysis of the different equilibrium points in the model is carried out. The basic reproduction number, R0, has been obtained too. The dependence of stability regions on the values of key parameters is demonstrated and the sensitivity of R0 to different parameters in the system is also investigated. Finally, a proposed optimal control scheme is adopted to successfully diminish the disease spread. It is found that the stability of disease-free equilibrium point is verified at small values of the infection rate of cattle caused by infected cattle or insects. In addition, the disease-free equilibrium point maintains its stability in cases where the recovery rate of cattle or the vaccination rate of cattle are increased. Moreover, the high-efficiency vaccination program helps in stabilizing the disease-free equilibrium point. Also, the fractional-order parameter is found to provide additional degree of freedom in the model to represent different scenarios of evolution for solution orbits. The optimal control parameters of the model are examined in different scenarios to limit or suppress the spread of the disease. It is demonstrated that two control parameters are sufficient to achieve the goals of disease elimination and stabilization of an infection-free steady state. Numerical simulations are verified to confirm our theoretical findings and estimations.

ANALYSIS AND SIMULATION OF THE SIR MODEL ON THE SPREAD OF COVID-19 BY CONSIDERING THE VACCINATION FACTOR

Covid-19 is a serious respiratory disease that can be fatal for those affected. Governments have tried various strategies to conquer the Covid-19 pandemic. One of them is to vaccinate people with 6 years old and over. The vaccination program aims to form herd immunity so that the number of confirmed positive cases can be reduced. The purpose of this research is to form a mathematical model of the SIR (Susceptible-Infected-Recovery) spread of Covid-19 by considering vaccination factors. The SIR model is combined with a vaccination factor to forestall the unfold of Covid-19. The research method includes deriving models of nonlinear differential equation systems, solving qualitative models, deriving the basic reproduction ratio ( ), analysis of equilibrium points, and building simulation models. This model has an asymptotically stable disease-free equilibrium point. At the same time, the endemic equilibrium point is unstable. Model simulation is obtained by using different parameter values. This is proven through the outcomes of the model analysis vaccination coverage is a key parameter that can be controlled to reduce so that the pandemic ends soon.

Dynamic Analysis of a Mathematical Model of the Anti-Tumor Immune Response

This study discusses the dynamic analysis, the Hopf bifurcation, and numerical simulations. The mathematical model of the anti-tumor immune response consists of three compartments namely Immature T Lymphocytes (L1), Mature T Lymphocytes (L2) and Tumor Cells (T). This research was conducted to represent the behavior between immune cells and tumor cells in the body with five γ conditions. Where γ is the intrinsic growth rate of mature T lymphocytes. This study produces R0 > 1 in conditions 1 to 4 while in condition 5 produces R0 < 1. The disease-free equilibrium point is stable only in condition 5, while the endemic equilibrium point is stable only in conditions 2 and 4. Hopf bifurcation occurs at the endemic equilibrium point. Numerical simulation graph in condition 1 shows that tumor cells will increase uncontrollably. In condition 2 the graph show that the endemic equilibrium point for large tumors is stable. In condition 3 the graph show that there will be a bifurcation from the endemic equilibrium point by the disturbance of the parameter value γ. In condition 4 the graph show the small tumor endemic equilibrium point is stable. Finally, in condition 5, the graph show a stable disease-free equilibrium point.

Modeling self-propagating malware with epidemiological models

Self-propagating malware (SPM) is responsible for large financial losses and major data breaches with devastating social impacts that cannot be understated. Well-known campaigns such as WannaCry and Colonial Pipeline have been able to propagate rapidly on the Internet and cause widespread service disruptions. To date, the propagation behavior of SPM is still not well understood. As result, our ability to defend against these cyber threats is still limited. Here, we address this gap by performing a comprehensive analysis of a newly proposed epidemiological-inspired model for SPM propagation, the Susceptible-Infected-Infected Dormant-Recovered (SIIDR) model. We perform a theoretical analysis of the SIIDR model by deriving its basic reproduction number and studying the stability of its disease-free equilibrium points in a homogeneous mixed system. We also characterize the SIIDR model on arbitrary graphs and discuss the conditions for stability of disease-free equilibrium points. We obtain access to 15 WannaCry attack traces generated under various conditions, derive the model’s transition rates, and show that SIIDR fits the real data well. We find that the SIIDR model outperforms more established compartmental models from epidemiology, such as SI, SIS, and SIR, at modeling SPM propagation.

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.

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.

Optimal control strategies to combat COVID-19 transmission: A mathematical model with incubation time delay

The coronavirus disease 2019, started spreading around December 2019, still persists in the population all across the globe. Though different countries have been able to cope with the disease to some extent and vaccination for the same has been developed, it cannot be ignored that the disease is still not on the verge of completely eradicating, which in turn creates a need for having deeper insights of the disease in order to understand it well and hence be able to work towards its eradication. Meanwhile, using mitigation strategies like non-pharmaceutical interventions can help in controlling the disease. In this work, our aim is to study the dynamics of COVID-19 using compartmental approach by applying various analytical methods. We obtain formula for important tools like R0 and establish the stability of disease-free equilibrium point for R0<1. Further, based on R0, we discuss the stability and existence of the endemic equilibrium point. We incorporate various control strategies possible and using optimal control theory, study their expected positive impacts on the spread of the disease. Later, using a biologically feasible set of parameters, we numerically analyse the model. We even study the trend of the outbreak in China, for over 120 days, where the active cases rise up to a peak and then the curve flattens.

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.

The Stability Analysis and Transmission Dynamics of the SIR Model with Nonlinear Recovery and Incidence Rates

In the present paper, the SIR model with nonlinear recovery and Monod type equation as incidence rates is proposed and analyzed. The expression for basic reproduction number is obtained which plays a main role in the stability of disease-free and endemic equilibria. The nonstandard finite difference (NSFD) scheme is constructed for the model and the denominator function is chosen such that the suggested scheme ensures solutions boundedness. It is shown that the NSFD scheme does not depend on the step size and gives better results in all respects. To prove the local stability of disease-free equilibrium point, the Jacobean method is used; however, Schur–Cohn conditions are applied to discuss the local stability of the endemic equilibrium point for the discrete NSFD scheme. The Enatsu criterion and Lyapunov function are employed to prove the global stability of disease-free and endemic equilibria. Numerical simulations are also presented to discuss the advantages of NSFD scheme as well as to strengthen the theoretical results. Numerical simulations specify that the NSFD scheme preserves the important properties of the continuous model. Consequently, they can produce estimates which are entirely according to the solutions of the model.

Optimal Control and Stability Analysis of an SEIR Model with Infectious Force in Latent Period.

In this paper, an SEWIR epidemic model with the government control rate and infectious force in latent period is proposed. The conditions to the existence and uniqueness of disease-free and endemic equilibrium points in the SEWIR model are obtained. By using the Hurwitz criterion, the locally asymptotic stability of disease-free and endemic equilibrium points is proved. We show the global asymptotic stability of the disease-free equilibrium point by the construction of Lyapunov function and LaSalle invariance principle. The globally asymptotic stability of the endemic equilibrium is verified by numerical simulation. Several optimal control strategies are proposed on controlling infectious diseases.

New Variant of SARS-CoV-2 Dynamics with Imperfect Vaccine

The occurrence of a new strain of SARS-CoV-2 cannot be ruled out. Therefore, this study seeks to investigate the possible effects of a hypothetical imperfect anti-COVID-19 vaccine on the control of not only the first variant of SARS-CoV-2 but also the second (new) variant of SARS-CoV-2. We further examine the rates r and a , escape of quarantined infectious individuals from isolation centers. The control R c and basic reproduction numbers R 0 are computed which gives assess to obtain asymptotic stability of disease-free equilibrium point globally and the existence of a unique persistent equilibrium solution. Numerical results reveal that people infected with the second strain who are vaccinated with an imperfect vaccine are under control but the prevalence of the second variant enhances the prevalence of the first variant. Thus, discovering a vaccine that is effective (to a good extent) for the prevention of variant 2 (new variant) is necessary for the control of COVID-19. Numerical results also reveal that increase in the rate at which individuals infected with the first variant escape the isolation center gives rise to the population infected with the first variant and lowers the peak of the population infected with the second variant. This is probably because individuals infected with the second variant appear to be more careful with their lives and get vaccinated more than individuals infected with the first variant.

Mathematical Model and Analysis on the Impact of Awareness Campaign and Asymptomatic Human Immigrants in the Transmission of COVID-19.

In this study, an autonomous type deterministic nonlinear mathematical model that explains the transmission dynamics of COVID-19 is proposed and analyzed by considering awareness campaign between humans and infectives of COVID-19 asymptomatic human immigrants. Unlike some of other previous model studies about this disease, we have taken into account the impact of awareness c between humans and infectives of COVID-19 asymptomatic human immigrants on COVID-19 transmission. The existence and uniqueness of model solutions are proved using the fundamental existence and uniqueness theorem. We also showed positivity and the invariant region of the model system with initial conditions in a certain meaningful set. The model exhibits two equilibria: disease (COVID-19) free and COVID-19 persistent equilibrium points and also the basic reproduction number, R0 which is derived via the help of next generation approach. Our analytical analysis showed that disease-free equilibrium point is obtained only in the absence of asymptomatic COVID-19 human immigrants and disease (COVID-19) in the population. Moreover, local stability of disease-free equilibrium point is verified via the help of Jacobian and Hurwitz criteria, and the global stability is verified using Castillo-Chavez and Song approach. The disease-free equilibrium point is both locally and globally asymptotically stable whenever R0 < 1, so that disease dies out in the population. If R0 > 1, then disease-free equilibrium point is unstable while the endemic equilibrium point exists and stable, which implies the disease persist and reinvasion will occur within a population. Furthermore, sensitivity analysis of the basic reproduction number, R0 with respect to model parameters, is computed to identify the most influential parameters in transmission as well as in the control of COVID-19. Finally, some numerical simulations are illustrated to verify the theoretical results of the model.

Modelling the impact of health care providers in transmission dynamics of COVID-19

In this paper, a mathematical model is proposed and analysed to assess the impacts of health care providers in transmission dynamics of COVID-19. The stability theory of differential equations is used to examine a mathematical model. The results of both local and global stability of disease-free equilibrium points were determined by using Routh–Hurwitz criteria and Metzler matrix method which verified that was locally and globally asymptotically stable. Also, the endemic equilibrium point was determined by the Lyapunov function which showed that E∗ was globally asymptotically stable under strict conditions. The findings revealed that non-diagnosed and undetected health care providers seems to contribute to high spread of COVID-19 in a community. Also, it illustrates that an increase in the number of non-diagnostic testing rates of health care providers may result in high infection rates in the community and contaminations of hospitals’ equipment. Therefore, the particular study recommend that there is a necessity of applying early diagnostic testing to curtail the COVID-19 transmission in the health care providers’ community and reduce contaminations of hospital’s equipment.

Modelling the dynamics of Cassava Mosaic Disease with non-cassava host plants

Cassava Mosaic Disease (CMD) is the prominent cassava disease which compromises cassava production in Africa, both qualitatively and quantitatively for many years. In this study, the mathematical model for the dynamics of CMD with Non-cassava host plant population is formulated and analysed. The analysis of basic model properties confirmed the positive boundedness of the model solution for all time t≥0. Utilizing the next generation matrix the basic reproduction number R0 is derived and stability of disease-free equilibrium point (DFE) is analysed. Analytical results confirmed that, the disease-free equilibrium point (DFE) is locally asymptotically stable whenever R0<1 and unstable otherwise. The sensitivity index analysis identified mortality rate and the carrying capacity of whiteflies as the most sensitive parameters of model. This implies that, any deliberate efforts towards controlling CMD should be directed into reducing the number of whiteflies. This can be implemented by either increasing the mortality rate of whiteflies or reducing their carrying capacity. In view of this, it is clear that deliberate measures such as the use of pesticides and entomopathogenic fungi which increases the parameter value of ω, and removing non-cassava host plants which will eventually decrease the parameter value of κw may bring significant results in combating CMD compared to the control of other model parameters. The decision regarding the best approach out of the two requires optimal control and cost-effective analysis of available control strategies. Furthermore, the numerical simulation results suggested that Cassava Mosaic Disease (CMD) can be controlled by increasing mortality rate of whiteflies, and the decreasing in the carrying capacity of whiteflies.

Stability analysis and backward bifurcation on an SEIQR epidemic model with nonlinear innate immunity

&lt;abstract&gt;&lt;p&gt;Infectious diseases have a great impact on the economy and society. Dynamic models of infectious diseases are an effective tool for revealing the laws of disease transmission. Quarantine and nonlinear innate immunity are the crucial factors in the control of infectious diseases. Currently, there no mathematical models that comprehensively study the effect of both innate immunity and quarantine. In this paper, we propose and analyze an SEIQR epidemic model with nonlinear innate immunity. The boundedness and positivity of the solutions are discussed. Employing the next-generation matrix, we compute the expression of the basic reproduction number. Under certain conditions, the phenomenon of backward bifurcation may occur. That is to say, the stable disease-free equilibrium point and the stable endemic equilibrium point coexist when the basic reproduction ratio is less than one. And the basic reproduction number is no longer the threshold value to determine whether the disease breaks out. We investigate the globally asymptotical stability of the disease-free equilibrium point for the system by constructing Lyapunov function. Also, we research the global stability of the endemic equilibrium by using geometric approach. Numerical simulations are carried out to reveal the theoretical results and find some complex dynamics (for example, the existence of Hopf bifurcation) of the system. Both theoretical and numerical results indicate that the nonlinear innate immunity may cause backward bifurcation and Hopf bifurcation, which makes more difficult to eliminate the disease.&lt;/p&gt;&lt;/abstract&gt;

Model matematika penyebaran COVID-19 dengan penggunaan masker kesehatan dan karantina

This study developed a model for the spread of COVID-19 disease using the SIR model which was added by a health mask and quarantine for infected individuals. The population is divided into six subpopulations, namely the subpopulation susceptible without a health mask, susceptible using a health mask, infected without using a health mask, infected using a health mask, quarantine for infected individuals, and the subpopulation to recover. The results obtained two equilibrium points, namely the disease-free equilibrium point and the endemic equilibrium point, and the basic reproduction number (R0). The existence of a disease-free equilibrium point is unconditional, whereas an endemic equilibrium point exists if the basic reproduction number is more than one. Stability analysis of the local asymptotically stable disease-free equilibrium point when the basic reproduction number is less than one. Furthermore, numerical simulations are carried out to provide a geometric picture related to the results that have been analyzed. The results of numerical simulations support the results of the analysis obtained. Finally, the sensitivity analysis of the basic reproduction numbers carried out obtained four parameters that dominantly affect the basic reproduction number, namely the rate of contact of susceptible individuals with infection, the rate of health mask use, the rate of health mask release, and the rate of quarantine for infected individuals.

Stochastic model of the transmission dynamics of COVID-19 pandemic

In this paper, we formulate an SVITR deterministic model and extend it to a stochastic model by introducing intensity of stochastic factors and Brownian motion. Our basic qualitative analysis of both models includes the positivity of the solution, invariant region, disease-free equilibrium point, basic reproduction number, local and global stability of disease-free equilibrium point, endemic equilibrium point, and sensitivity. We obtain the stochastic reproduction number and local stability by using twice differentiable Itô’s formula. We prove the global stability of the disease-free equilibrium point by using a Lyapunov function. We determine the sensitivity of the effect of each parameter on basic reproduction number of the model by using a normalized sensitivity index formula. On the other hand, we demonstrate numerical simulation results of deterministic and stochastic models of COVID-19 by using Maple 18 and MATLAB software. Our simulation results indicate that reducing the contact between infected and susceptible individuals and improvement of treatment play a vital role in COVID-19 pandemic control.

A fractional order HIV-TB co-infection model in the presence of exogenous reinfection and recurrent TB.

In this article, a novel fractional order model has been introduced in Caputo sense for HIV-TB co-infection in the presence of exogenous reinfection and recurrent TB along with the treatment for both HIV and TB. The main aim of considering the fractional order model is to incorporate the memory effect of both diseases. We have analyzed both sub-models separately with fractional order. The basic reproduction number, which measures the contagiousness of the disease, is determined. The HIV sub-model is shown to have a locally asymptotically stable disease-free equilibrium point when the corresponding reproduction number, mathcal {R}_H, is less than unity, whereas, for mathcal {R}_H>1, the endemic equilibrium point comes into existence. For the TB sub-model, the disease-free equilibrium point has been proved to be locally asymptotically stable for mathcal {R}_T<1. The existence of TB endemic equilibrium points in the presence of reinfection and recurrent TB for mathcal {R}_T<1 justifies the existence of backward bifurcation under certain restrictions on the parameters. Further, we numerically simulate the fractional order model to verify the analytical results and highlight the role of fractional order in co-infection modeling. The fractional order derivative is shown to have a crucial role in determining the transmission dynamics of HIV-TB co-infection. It is concluded that the memory effect plays a significant role in reducing the infection prevalence of HIV-TB co-infection. An increment in the number of recovered individuals can also be observed when the memory effect is taken into consideration by introducing fractional order model.

A study on canine distemper virus (CDV) and rabies epidemics in the red fox population via fractional derivatives

Several deadly epidemics that have recognized as serious problems all over the world in the last few decades. Lassa hemorrhagic fever, coronavirus, dengue fever, malaria, and HIV are well-known deadly diseases in humans. In this research, we analysed the dynamics of the canine distemper virus (CDV) and rabies epidemics in the red fox population of the northern region of Italy with the help of time-fractional models. We performed our analysis in the new generalized Caputo non-classical derivative sense with the application of the Predictor–Corrector algorithm. We used the data of northern Italy for simulations and estimated the endemic equilibrium points for both CDV and rabies models. Also, we presented the local stability of disease-free equilibrium points. Some theorems are mentioned for the purpose of existence and uniqueness analysis. Our results are perfect for giving an idea of the dynamics of the CDV and rabies epidemic in northern Italy. The dynamics of the given solutions are specified with the help of necessary graphical simulations. The projected algorithm is so effective in finding the solutions of complex dynamical systems. By this study, we give an idea of how applied mathematics is directly connected to biological studies. The major scientific aim of this study is to understand the outbreaks of CDV and rabies on the population of the red foxes by using the texture of fractional mathematical models.

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.