Local Stability Of Disease-free Equilibrium Research Articles

Modeling the transmission dynamics of COVID-19 with genetically resistant humans

The emergence of a coronavirus disease (COVID-19) pandemic caused by severe acute respiratory syndrome coronavirus (SARS-COV 2) has caused an unprecedented global societal and public health crisis. A mathematical model that incorporates an individual’s ability to resist COVID-19 infection is considered to study the dynamics of COVID-19. Detection and testing for COVID-19 in individuals who has ability to resist the infection, proper preventive measures, and timely medical care are essential in managing and controlling COVID-19 outbreak. These allows us to make quantifiable predictions about the effects of testing utilization on the COVID-19 prevalence. First, we analyze the disease-free state of the COVID-19 model, calculate the baseline control reproduction number, and obtain the local stability of disease-free equilibrium. The model is calibrated using data obtained from Nigeria Centre for Disease Control and key parameters of the model are estimated. Sensitivity analysis is carried out to investigate the influence of the parameters in curtailing the disease. Numerical simulations are then used to explore the behavior of the model solutions involving the novel variable (genetically resistant humans) and control measures. Our results suggest that strict intervention effort (early detection of genetically resistant individuals, testing and isolation of the infected individuals) is required for quick suppression of the disease.

Mathematical Analysis of COVID-19 model with Vaccination and Partial Immunity to Reinfection

COVID-19 is an infectious respiratory disease caused by a new virus, called SARS-CoV-2. Since itsinception, it has been a major cause of deaths and illnesses in the general population across the globe. Inthis paper, we have formulated and theoretically analyzed a non-linear deterministic model for COVID-19transmission dynamics by incorporating vaccination of the susceptible population. The system properties,such as the boundedness of solutions, the basic reproduction number R0, the local stability of disease-freeequilibrium(DFE), and endemic equilibrium (EE) points, are explored. Besides, the Lyapunov function isutilized to prove the global stability of both DFE and EE. The bifurcation analysis was carried out by utilizingthe center manifold theory. Then, the model is fitted with real COVID-19 cumulative data of infected casesin Kenya as from March 30, 2020, to March 30, 2022. Furthermore, sensitivity analysis was performed forthe proposed model to ascertain the relative significance of model parameters to COVID-19 transmissiondynamics. The simulations revealed that the spread of COVID-19 can be curtailed not only via vaccinationof susceptible populations but also increased administration of COVID-19 booster vaccine to the vaccinatedpersons and early detection and treatment of asymptomatic individuals.

Stability of Covid-19 Dynamics: A Case Study of Nigeria

In this paper, a SEIR epidemic model is considered; where individuals in the population are assigned to different compartments of SEIR defined with respect to epidemic status of Covid-19 in Nigeria. The article has demonstrated a simple mathematical model for the transmission of Covid-19 disease taking into account loss of human immunity with the aim that this model proves useful in controlling the possibility of a person contracting Covid-19 twice. When the basic reproduction number means that the Covid-19 free equilibrium solution is locally asymptotically stable. This suggests that the number of new cases of the disease will decrease over time and eventually will vanish as that whcih causes are established. The basic reproduction number and the model analysis (local stability of disease-free equilibrium and disease-endemic equilibrium) of the system were calculated and the stability of the SEIR model was checked.

Why Controlling the Asymptomatic Infection Is Important: A Modelling Study with Stability and Sensitivity Analysis

The large proportion of asymptomatic patients is the major cause leading to the COVID-19 pandemic which is still a significant threat to the whole world. A six-dimensional ODE system (SEIAQR epidemical model) is established to study the dynamics of COVID-19 spreading considering infection by exposed, infected, and asymptomatic cases. The basic reproduction number derived from the model is more comprehensive including the contribution from the exposed, infected, and asymptomatic patients. For this more complex six-dimensional ODE system, we investigate the global and local stability of disease-free equilibrium, as well as the endemic equilibrium, whereas most studies overlooked asymptomatic infection or some other virus transmission features. In the sensitivity analysis, the parameters related to the asymptomatic play a significant role not only in the basic reproduction number R0. It is also found that the asymptomatic infection greatly affected the endemic equilibrium. Either in completely eradicating the disease or achieving a more realistic goal to reduce the COVID-19 cases in an endemic equilibrium, the importance of controlling the asymptomatic infection should be emphasized. The three-dimensional phase diagrams demonstrate the convergence point of the COVID-19 spreading under different initial conditions. In particular, massive infections will occur as shown in the phase diagram quantitatively in the case R0>1. Moreover, two four-dimensional contour maps of Rt are given varying with different parameters, which can offer better intuitive instructions on the control of the pandemic by adjusting policy-related parameters.

A mathematical analysis of the dynamics of chikungunya virus transmission

In this paper, a deterministic model for the dynamics of chikungunya virus transmission is formulated and analyzed. It is shown that the model has a disease free equilibrium (DFE) and by using the basic reprodution number (R0) local stability of DFE is proved when R0 < 1. Also, the global stability of DFE is investigated by Lyapunov function and LaSalle Invariance Principle. We show that there exists a unique endemic equilibrium (EE) of the model which is locally asymptotically stable whenever R0 > 1 and establish the global stability of the EE when R0 > 1, by using Lyapunov function and LaSalle Invariance Principle for a special case. Numerical simulations and sensitivity analysis show that the destruction of breeding sites and reduction of average life spans of vector would be effective prevention to control the outbreak. Controlling of effective contact rates and reducing transmissions probabilities may reduce the disease prevalence. GANITJ. Bangladesh Math. Soc.41.1 (2021) 41-61

Effects of Nonmonotonic Functional Responses on a Disease Transmission Model: Modeling and Simulation.

In this article, a novel susceptible–infected–recovered epidemic model with nonmonotonic incidence and treatment rates is proposed and analyzed mathematically. The Monod–Haldane functional response is considered for nonmonotonic behavior of both incidence rate and treatment rate. The model analysis shows that the model has two equilibria which are named as disease-free equilibrium (DFE) and endemic equilibrium (EE). The stability analysis has been performed for the local and global behavior of the DFE and EE. With the help of the basic reproduction number left( {R_{0} } right) , we investigate that DFE is locally asymptotically stable when R_{0} < 1 and unstable when R_{0} > 1 . The local stability of DFE at R_{0} = 1 has been analyzed, and it is obtained that DFE exhibits a forward transcritical bifurcation. Further, we identify conditions for the existence of EE and show the local stability of EE under certain conditions. Moreover, the global stability behavior of DFE and EE has been investigated. Lastly, numerical simulations have been done in the support of our theoretical findings.

Analysis of a delayed epidemic model with non-monotonic incidence rate and vertical transmission

A delayed SEIR epidemic model with vertical transmission and non-monotonic incidence is formulated. The equilibria and the threshold of the model have been determined on the bases of the basic reproduction number. The local stability of disease-free equilibrium and endemic equilibrium is established by analyzing the corresponding characteristic equations. By comparison arguments, it is proved that, if R0 &lt; 1, the disease-free equilibrium is globally asymptotically stable. Whereas, the disease-free equilibrium is unstable if R0 &gt; 1. Moreover, we show that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium when R0 &gt; 1.

Global dynamics of a network epidemic model for waterborne diseases spread

A network epidemic model for waterborne diseases spread is formulated, which incorporates both indirect environment-to-human and direct human-to-human transmission routes. We consider direct human contacts as a heterogeneous network and assume homogeneous mixing between the environment and human population. The basic reproduction number R0 is derived through the local stability of disease-free equilibrium and established as a sharp threshold that governs the disease dynamics. In particular, we have shown that the disease-free equilibrium is globally asymptotically stable if R0<1; while if R0>1, in case of permanent immunity the endemic equilibrium is globally asymptotically stable based on Kirchhoff’s matrix tree theorem, and in case of temporal immunity sufficient conditions are given to guarantee global stability of the endemic equilibrium. Moreover, various immunization strategies are investigated and compared. The results obtained are informative for us to further understand the disease propagation in multiple transmission routes and devise some effective intervening measures to fight against the diseases spread.

STABILITY ANALYSIS OF A DELAYED SIRS EPIDEMIC MODEL WITH VACCINATION AND NONLINEAR INCIDENCE

In this paper, an SIRS epidemic model with time delay and vaccination is investigated. By analyzing the corresponding characteristic equation, the local stability of disease-free equilibrium of the model is established. By constructing Lyapunov functional, sufficient conditions are established for the local stability of an endemic equilibrium of the model. Further, a threshold value is obtained. By using comparison arguments, it is proved when the threshold value is less than unity, the disease-free equilibrium is globally asymptotically stable. When the threshold value is greater than unity, by using an iteration scheme and by constructing appropriate Lyapunov functional, respectively, sufficient conditions are derived for the global stability of the endemic equilibrium of the model. Numerical simulations are carried out to illustrate the theoretical results.

Dynamical Analysis of Delayed Plant Disease Models with Continuous or Impulsive Cultural Control Strategies

Delayed plant disease mathematical models including continuous cultural control strategy and impulsive cultural control strategy are presented and investigated. Firstly, we consider continuous cultural control strategy in which continuous replanting of healthy plants is taken. The existence and local stability of disease‐free equilibrium and positive equilibrium are studied by analyzing the associated characteristic transcendental equation. And then, plant disease model with impulsive replanting of healthy plants is also considered; the sufficient condition under which the infected plant‐free periodic solution is globally attritive is obtained. Moreover, permanence of the system is studied. Some numerical simulations are also given to illustrate our results.

Epidemic spreading of an SEIRS model in scale-free networks

An SEIRS epidemic model on the scale-free networks is presented, where the active contact number of each vertex is assumed to be either constant or proportional to its degree for this model. Using the analytical method, we obtain the two threshold values for above two cases and find that the threshold value for constant contact is independent of the topology of the underlying networks. The existence of positive equilibrium is determined by threshold value. For a finite size of scale-free network, we prove the local stability of disease-free equilibrium and the permanence of the disease on the network. Furthermore, we investigate two major immunization strategies, random immunization and targeted immunization, some similar results are obtained. The simulation shows the positive equilibrium is stable.