scholarly journals A non-standard discretized SIS model of epidemics

2021 ◽  
Vol 19 (1) ◽  
pp. 115-133
Author(s):  
Marcin Choiński ◽  
◽  
Mariusz Bodzioch ◽  
Urszula Foryś ◽  
◽  
...  
Keyword(s):  
Global Stability ◽  
Stationary State ◽  
Continuous System ◽  
Continuous Model ◽  
Free State ◽  
Discrete Version ◽  
Step Size ◽  
Sis Model ◽  
Homogeneous Population ◽  
Disease Free

<abstract><p>In this paper we introduce and analyze a non-standard discretized SIS epidemic model for a homogeneous population. The presented model is a discrete version of the continuous model known from literature and used by us for building a model for a heterogeneous population. Firstly, we discuss basic properties of the discrete system. In particular, boundedness of variables and positivity of solutions of the system are investigated. Then we focus on stability of stationary states. Results for the disease-free stationary state are depicted with the use of a basic reproduction number computed for the system. For this state we also manage to prove its global stability for a given condition. It transpires that the behavior of the disease-free state is the same as its behavior in the analogous continuous system. In case of the endemic stationary state, however, the results are presented with respect to a step size of discretization. Local stability of this state is guaranteed for a sufficiently small critical value of the step size. We also conduct numerical simulations confirming theoretical results about boundedness of variables and global stability of the disease-free state of the analyzed system. Furthermore, the simulations ascertain a possibility of appearance of Neimark-Sacker bifurcation for the endemic state. As a bifurcation parameter the step size of discretization is chosen. The simulations suggest the appearance of a supercritical bifurcation.</p></abstract>

Convergence and Monotonicity for a Model of Spontaneous Infection and Transmission

2014 ◽  
Vol 46 (02) ◽  
pp. 560-584
Author(s):  
Eric Foxall
Keyword(s):  
Initial Data ◽  
Exponential Convergence ◽  
Contact Process ◽  
Free State ◽  
Sis Model ◽  
Spontaneous Process ◽  
Finite Set ◽  
Disease Free ◽  
Spontaneous Infection

A version of the contact process (effectively an SIS model) on a finite set of sites is considered in which there is the possibility of spontaneous infection. A companion process is also considered in which spontaneous infection does not occur from the disease-free state. Monotonicity with respect to parameters and initial data is established, and conditions for irreducibility and exponential convergence of the processes are given. For the spontaneous process, a set of approximating equations is derived, and its properties investigated.

scholarly journals On Global Stability of Disease-Free Equilibrium in Epidemiological Models

2021 ◽  
Vol 2 (3) ◽  
pp. 37-42
Author(s):  
Tunde Tajudeen Yusuf
Keyword(s):  
Lyapunov Function ◽  
Global Stability ◽  
Direct Approach ◽  
Epidemiological Models ◽  
Homogeneous Population ◽  
Disease Free

This paper considers the problem of constructing appropriate Lyapunov function for establishing the global stability of a disease-free equilibrium in epidemiological models. A generalised algorithm is proposed and it is tested for some selected epidemiological models. Experience from the application of the algorithm on test examples shows that the algorithm is easy to use, less cumbersome, and yielded the desired result, particularly in models with homogeneous population. Thus, the proposed algorithm provides a direct approach for establishing global stability of disease-free equilibrium.

scholarly journals A Consistent Discrete Version of a Nonautonomous SIRVS Model

2018 ◽  
Vol 2018 ◽  
pp. 1-18
Author(s):  
Joaquim Mateus ◽  
César M. Silva ◽  
S. Vaz
Keyword(s):  
Discrete Model ◽  
Real Data ◽  
Continuous Model ◽  
Discrete Models ◽  
Discrete Version ◽  
Continuous Family ◽  
Discretization Method ◽  
Time Step ◽  
The Stability ◽  
Disease Free

A family of discrete nonautonomous SIRVS models with general incidence is obtained from a continuous family of models by applying Mickens nonstandard discretization method. Conditions for the permanence and extinction of the disease and the stability of disease-free solutions are determined. Concerning extinction and persistence, the consistency of those discrete models with the corresponding continuous model is discussed: if the time step is sufficiently small, when we have extinction (permanence) for the continuous model, we also have extinction (permanence) for the corresponding discrete model. Some numerical simulations are carried out to compare the different possible discretizations of our continuous model using real data.

Convergence and Monotonicity for a Model of Spontaneous Infection and Transmission

2014 ◽  
Vol 46 (2) ◽  
pp. 560-584
Author(s):  
Eric Foxall
Keyword(s):  
Initial Data ◽  
Exponential Convergence ◽  
Contact Process ◽  
Free State ◽  
Sis Model ◽  
Spontaneous Process ◽  
Finite Set ◽  
Disease Free ◽  
Spontaneous Infection

A version of the contact process (effectively an SIS model) on a finite set of sites is considered in which there is the possibility of spontaneous infection. A companion process is also considered in which spontaneous infection does not occur from the disease-free state. Monotonicity with respect to parameters and initial data is established, and conditions for irreducibility and exponential convergence of the processes are given. For the spontaneous process, a set of approximating equations is derived, and its properties investigated.

EPIDEMIC SPREADING AND GLOBAL STABILITY OF A NEW SIS MODEL WITH DELAY ON HETEROGENEOUS NETWORKS

2015 ◽  
Vol 23 (04) ◽  
pp. 1550029 ◽  
Author(s):  
HUIYAN KANG ◽  
YIJUN LOU ◽  
GUANRONG CHEN ◽  
SEN CHU ◽  
XINCHU FU
Keyword(s):  
Global Stability ◽  
Heterogeneous Networks ◽  
Lyapunov Functional ◽  
Functional Differential Equations ◽  
Epidemic Spreading ◽  
Epidemic Threshold ◽  
Sis Model ◽  
Vector Population ◽  
Functional Differential ◽  
Disease Free

In this paper, we study a susceptible-infected-susceptible (SIS) model with time delay on complex heterogeneous networks. Here, the delay describes the incubation period in the vector population. We calculate the epidemic threshold by using a Lyapunov functional and some analytical methods, and find that adding delay increases the epidemic threshold. Then, we prove the global stability of disease-free and endemic equilibria by using the theory of functional differential equations. Furthermore, we show numerically that the epidemic threshold of the new model may change along with other factors, such as the infectivity function, the heterogeneity of the network, and the degrees of nodes. Finally, we find numerically that the delay can affect the convergence speed at which the disease reaches equilibria.

scholarly journals A Discrete Numerical Solution of The SIR Model with Horizontal and Vertical Transmission

CAUCHY ◽  
2019 ◽  
Vol 6 (1) ◽  
pp. 1
Author(s):  
Trija Fayeldi
Keyword(s):  
Numerical Solution ◽  
Vertical Transmission ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Sir Model ◽  
Euler Method ◽  
Discrete Version ◽  
Step Size ◽  
The Stability ◽  
Disease Free

The aim of this paper is to is to generalize the SIR model with horizontal and vertical transmission. In this paper, we develop the discrete version of the model. We use Euler method to approximate numerical solution of the model. We found two equilibrium points, that is disease free and endemic equilibrium points. The existence of these points depend on basic reproduction number <em>R</em><sub>0</sub>. We found that if <em>R</em><sub>0</sub> <span style="text-decoration-line: underline;">&lt;</span> 1 then only disease free equilibrium points exists, while both points exists when <em>R</em><sub>0</sub> &gt; 1. We also found that the stability of these equilibrium points depend on the value of step-size <em>h</em>. Some numerical experiments were presented as illustration.

scholarly journals Volterra–Lyapunov Stability Analysis of the Solutions of Babesiosis Disease Model

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1272
Author(s):  
Fengsheng Chien ◽  
Stanford Shateyi
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Global Stability ◽  
Numerical Simulations ◽  
Lyapunov Stability ◽  
Disease Model ◽  
Transmission Dynamics ◽  
Global Stability Analysis ◽  
Lyapunov Stability Analysis ◽  
Disease Free

This paper studies the global stability analysis of a mathematical model on Babesiosis transmission dynamics on bovines and ticks populations as proposed by Dang et al. First, the global stability analysis of disease-free equilibrium (DFE) is presented. Furthermore, using the properties of Volterra–Lyapunov matrices, we show that it is possible to prove the global stability of the endemic equilibrium. The property of symmetry in the structure of Volterra–Lyapunov matrices plays an important role in achieving this goal. Furthermore, numerical simulations are used to verify the result presented.

scholarly journals A Mass- and Energy-Conserving Numerical Model for a Fractional Gross–Pitaevskii System in Multiple Dimensions

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1765
Author(s):  
Adán J. Serna-Reyes ◽  
Jorge E. Macías-Díaz
Keyword(s):  
Initial Conditions ◽  
Type System ◽  
Continuous System ◽  
Mass Conservation ◽  
Continuous Model ◽  
Manuscript Studies ◽  
Multiple Dimensions ◽  
Negative Constant ◽  
Energy Conserving ◽  
Finite Difference Discretization

This manuscript studies a double fractional extended p-dimensional coupled Gross–Pitaevskii-type system. This system consists of two parabolic partial differential equations with equal interaction constants, coupling terms, and spatial derivatives of the Riesz type. Associated with the mathematical model, there are energy and non-negative mass functions which are conserved throughout time. Motivated by this fact, we propose a finite-difference discretization of the double fractional Gross–Pitaevskii system which inherits the energy and mass conservation properties. As the continuous model, the mass is a non-negative constant and the solutions are bounded under suitable numerical parameter assumptions. We prove rigorously the existence of solutions for any set of initial conditions. As in the continuous system, the discretization has a discrete Hamiltonian associated. The method is implicit, multi-consistent, stable and quadratically convergent. Finally, we implemented the scheme computationally to confirm the validity of the mass and energy conservation properties, obtaining satisfactory results.

scholarly journals Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Yanju Xiao ◽  
Weipeng Zhang ◽  
Guifeng Deng ◽  
Zhehua Liu
Keyword(s):  
Sufficient Conditions ◽  
Global Dynamics ◽  
Sis Model ◽  
Delayed Treatment ◽  
Sis Epidemic Model ◽  
Treatment Function ◽  
Lyapunov Coefficient ◽  
Saturated Treatment ◽  
Theoretical Results ◽  
Disease Free

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.

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