Dynamics of a two-patch epidemic model with deterministic/stochastic migration and distributed delays.

In this paper, a two-patch SIS model with saturating contact rate and one-directing population dispersal is proposed. In the model, individuals can only migrate from patch 1 to patch 2. The basic reproduction number $ R_0^1 $ of patch 1 and the basic reproduction number $ R_0^2 $ of patch 2 is identified. The global dynamics are completely determined by the two reproduction numbers. It is shown that if $ R_0^1 < 1 $ and $ R_0^2 < 1 $, the disease-free equilibrium is globally asymptotically stable; if $ R_0^1 < 1 $ and $ R_0^2 > 1 $, there is a boundary equilibrium which is globally asymptotically stable; if $ R_0^1 > 1 $, there is a unique endemic equilibrium which is globally asymptotically stable. Finally, numerical simulations are performed to validate the theoretical results and reveal the influence of saturating contact rate and migration rate on basic reproduction number and the transmission scale.

A weighted networked SIRS epidemic model

In this paper, we study an SIRS epidemic model with a nonlinear incidence rate and vaccination. We give the existence of positivity, and boundedness of the equilibrium of the model. We calculate the basic reproduction number of the proposed model by using the next generation matrix method. By constructing Lyapunov function, we show that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less or equal than one and that the endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than one. Numerical simulations are performed to investigate the effect of vaccinate on model behavior.

Rotavirus is the leading cause of severe diarrhea in children worldwide. To study the spread of rotavirus, a rotavirus transmission model was proposed based on the characteristics of temporary immunity after infection and maternal antibody protecting the newborn. By means of dynamic analysis, the basic reproduction number deciding the persistence of the infection was obtained. Based on the local stability analysis of the feasible equilibria, it was proved that the diseasefree equilibrium will be globally asymptotically stable if the basic reproduction number is no more than 1, through construction of appropriate Lyapunov functions. The disease will persist in the population if the basic reproduction number is more than 1 according to the Fonda lemma.

Mathematical perspective of Covid-19 pandemic: Disease extinction criteria in deterministic and stochastic models.

We proposed a deterministic compartmental model for the transmission dynamics of COVID-19 disease. We performed qualitative and quantitative analysis of the deterministic model concerning the local and global stability of the disease-free and endemic equilibrium points. We found that the disease-free equilibrium is locally asymptotically stable when the basic reproduction number is less than unity, while the endemic equilibrium point becomes locally asymptotically stable if the basic reproduction number is above unity. Furthermore, we derived the global stability of both the disease-free and endemic equilibriums of the system by constructing some Lyapunov functions. If R 0 ≤ 1 , it is found that the disease-free equilibrium is globally asymptotically stable, while the endemic equilibrium point is globally asymptotically stable when R 0 &gt; 1 . The numerical results of the general dynamics are in agreement with the theoretical solutions. We established the optimal control strategy by using Pontryagin’s maximum principle. We performed numerical simulations of the optimal control system to investigate the impact of implementing different combinations of optimal controls in controlling and eradicating COVID-19 disease. From this, a significant difference in the number of cases with and without controls was observed. We observed that the implementation of the combination of the control treatment rate, u 2 , and the control treatment rate, u 3 , has shown effective and efficient results in eradicating COVID-19 disease in the community relative to the other strategies.

In this paper, the probability properties are investigated for a stochastic SIS epidemic model. Transition probabilities of the susceptible process are obtained by using Laplace transform and perturbation variables. According to two cases: basic reproduction number [Formula: see text] and [Formula: see text], the dynamical behaviors in probability of the process are analyzed. It is shown that when [Formula: see text] the disease-free equilibrium is globally asymptotically stable with probability one, and when [Formula: see text] and [Formula: see text] is a positive integer, the endemic equilibrium is globally asymptotically stable with probability one. These results coincide with the corresponding deterministic SIS epidemic model. However, when [Formula: see text] and [Formula: see text] is not a positive integer, there are different properties between the deterministic and stochastic models. Numerical simulations are also performed to validate these results.

A two-patch model,SEi1,…,EinIiLi, i=1,2, is used to analyze the spread of tuberculosis, with an arbitrary numbernof latently infected compartments in each patch. A fraction of infectious individuals that begun their treatment will not return to the hospital for the examination of sputum. This fact usually occurs in sub-Saharan Africa, due to many reasons. The model incorporates migrations from one patch to another. The existence and uniqueness of the associated equilibria are discussed. A Lyapunov function is used to show that when the basic reproduction ratio is less than one, the disease-free equilibrium is globally and asymptotically stable. When it is greater than one, there exists at least one endemic equilibrium. The local stability of endemic equilibria can be illustrated using numerical simulations. Numerical simulation results are provided to illustrate the theoretical results and analyze the influence of lost sight individuals.

In this paper, by constructing Lyapunov functionals, we consider the global dynamics of an SIRS epidemic model with a wide class of nonlinear incidence rates and distributed delays \(\int^{h}_{0} p(\tau)f(S(t),I(t-\tau)) \mathrm{d}\tau\) under the condition that the total population converges to 1. By using a technical lemma which is derived from strong condition of strict monotonicity of functions f(S,I) and f(S,I)/I with respect to S≥0 and I>0, we extend the global stability result for an SIR epidemic model if R0>1, where R0 is the basic reproduction number. By using a limit system of the model, we also show that the disease-free equilibrium is globally asymptotically stable if R0=1.

Strangles is one of the most prevalent horse diseases globally. The infected horses may be asymptomatic and can still carry the infectious pathogen after it recovers, which are named asymptomatic infected horses and long-term subclinical carriers, respectively. Based on these horses, this paper establishes a dynamical model to screen, measure, and model the spread of strangles. The basic reproduction number $ \mathcal{R}_0 $ is computed through a next generation matrix method. By constructing Lyapunov functions, we concluded that the disease-free equilibrium is globally asymptotically stable if $ \mathcal{R}_0 < 1 $, and the endemic equilibrium exits uniquely and is globally asymptotically stable if $ \mathcal{R}_0 > 1 $. For example, while studying a strangles outbreak of a horse farm in England in 2012, we computed an $ \mathcal{R}_0 = 0.8416 $ of this outbreak by data fitting. We further conducted a parameter sensitivity analysis of $ \mathcal{R}_0 $ and the final size by numerical simulations. The results show that the asymptomatic horses mainly influence the final size of this outbreak and that long-term carriers are connected to an increased recurrence of strangles. Moreover, in terms of the three control measures implemented to control strangles(i.e., vaccination, implementing screening regularly and isolating symptomatic horses), the result shows that screening is the most effective measurement, followed by vaccination and isolation, which can provide effective guidance for horse management.

Purpose: This research is about a new COVID-19 SIR model containing three classes; susceptible S(t), infected I(t), and recovered R(t) with the convex incident rate. Methodology: The NCOVID-19 model was formulated in the following system, the whole population N(t) was divided into three classes S(t), I(t), and R(t), which represented Susceptible, Infected, and Recovered compartments in the form of differential equations. Lyapunov functions were used to validate the stability of the equilibrium of the ordinary differential equations, linearization of the system was also done using Jacobian matrices by finding the derivatives of f(x) for x. Findings: Covid-19 is an infectious disease caused by the novel coronavirus identified as Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2). The people infected by COVID-19 experience mild respiratory problems such as; Fever, dry cough, throat infection, and fatigue. People may also have symptoms such as nasal infection, aches, and sore throat. The pandemic has led to a dramatic loss of human life in Kenya, Africa, and the whole world as it presents an unprecedented challenge to public health, food systems, and the world of work. This case study seeks to model covid-19 virus after lifting preventive measures with a major focus on Kisii County, the subject model was presented in the form of differential equations and the disease-free and endemic equilibrium was calculated for the model. Also, the basic reproduction number R0 = 0.7831 was calculated and the disease-free equilibrium was found to be asymptotically stable meaning that the virus could be eliminated from the population, this showed that the county government of Kisii was in good control of the COVID-19 situation., in addition, The global stability of the model was calculated using the Lyapunov function construction while the Local stability was calculated using the Jacobian matrices. The numerical solutions were calculated using the non-standard finite difference scheme (NFDS) and MATLAB software. Unique Contribution to Theory, Practice and Policy: This study has laid a foundation for future research in the area. In the future, a study that can include the rate of COVID-19 virus mutation and its impacts is recommended.

In this paper, we present and analyze a SVIR epidemic mathematical model for rotavirus infection with vaccination and saturated incidence rate. Dynamical analysis of this model is done by determining the equilibrium point and stability of the equibrilium point. The model exhibits two equilibrium points, i.e. disease free and endemic equilibrium. The basic reproduction number Rv and R 0 has been obtained. The stability of the disease free and endemic equilibrium exists when the basic reproduction number less or greater than unity, respectively. Analytical result shows that those equilibrium points are locally asymptotically stable under certain condition. It is proved that the disease free equilibrium is globally stable when the value of basic reproduction number Rv < 1 and R 0 < 1, respectively. Numerical simulations are presented to support and complement the theoretical results.

The basic reproduction number, ℛ0, one of the most well-known thresholds in deterministic epidemic theory, predicts a disease outbreak if ℛ0>1. In stochastic epidemic theory, there are also thresholds that predict a major outbreak. In the case of a single infectious group, if ℛ0>1 and i infectious individuals are introduced into a susceptible population, then the probability of a major outbreak is approximately 1−(1/ℛ0) i . With multiple infectious groups from which the disease could emerge, this result no longer holds. Stochastic thresholds for multiple groups depend on the number of individuals within each group, i j , j=1, …, n, and on the probability of disease extinction for each group, q j . It follows from multitype branching processes that the probability of a major outbreak is approximately . In this investigation, we summarize some of the deterministic and stochastic threshold theory, illustrate how to calculate the stochastic thresholds, and derive some new relationships between the deterministic and stochastic thresholds.

A mathematical model for the transmission of cholera dynamics with a class of quarantined and vaccination parameter as control strategies is proposed in this paper. It is shown through mathematical analysis that the solution of the model uniquely exist, is positive and bounded in a certain region. The disease-free and endemic equilibrium points of the model are obtained. By using the next generation matrix, the basic reproduction number was computed around the disease-free equilibrium points, and it was shown through the Jacobian matrix that the disease free equilibrium is locally asymptotic stable if Rh<1. Numerical simulation was carried to understand the impact of the incorporated controls as the system evolves over time. Results show that effective quarantine, vaccination and proper sanitation reduce the disease contact rates and thus eliminates the spread of cholera.

In this paper, an SEIVR epidemic model with generalized incidence and preventive vaccination is considered. First, we formulate the model and obtain its basic properties. Then, we find the equilibrium points of the model, the disease-free and the endemic equilibrium. The stability of disease-free and endemic equilibrium is associated with the basic reproduction number [Formula: see text]. If the basic reproduction number [Formula: see text], the disease-free equilibrium is locally as well as globally asymptotically stable. Moreover, if the basic reproduction number [Formula: see text], the disease is uniformly persistent and the unique endemic equilibrium of the system is locally as well as globally asymptotically stable under certain conditions. Finally, the numerical results justify the analytical results.