Pulse vaccination in the periodic infection rate SIR epidemic model

A pulse vaccination SIR model with periodic infection rate $\beta (t)$ have been proposed and studied. The basic reproductive number $R_0$ is defined. The dynamical behaviors of the model are analyzed with the help of persistence, bifurcation and global stability. It has been shown that the infection-free periodic solution is globally stable provided $R_0 < 1$ and is unstable if $R_0>1$. Standard bifurcation theory have been used to show the existence of the positive periodic solution for the case of $R_0 \to1^+$. Finally, the numerical simulations have been performed to show the uniqueness and the global stability of the positive periodic solution of the system.Comment: 17pages and 3figures, submmission to Mathematical Bioscience

Impulsive Vaccination SEIR Model with Nonlinear Incidence Rate and Time Delay

This paper aims to discuss the delay epidemic model with vertical transmission, constant input, and nonlinear incidence. Some sufficient conditions are given to guarantee the existence and global attractiveness of the infection-free periodic solution and the uniform persistence of the addressed model with time delay. Finally, a numerical example is given to demonstrate the effectiveness of the proposed results

Stability analysis of drinking epidemic models and investigation of optimal treatment strategy

In this research we investigate a class of drinking epidemic models, namely the SPARS type models. The basic reproduction number is derived, and the system dynamical behaviours are investigated for both drinking free equilibrium and drinking persistent equilibrium. The purpose is to determine the long term optimal treatment method and the optimal short period vaccination strategy for controlling the population of the periodic drinkers and alcoholics

SVEIRS: A New Epidemic Disease Model with Time Delays and Impulsive Effects

We first propose a new epidemic disease model governed by system of impulsive delay differential equations. Then, based on theories for impulsive delay differential equations, we skillfully solve the difficulty in analyzing the global dynamical behavior of the model with pulse vaccination and impulsive population input effects at two different periodic moments. We prove the existence and global attractivity of the “infection-free” periodic solution and also the permanence of the model. We then carry out numerical simulations to illustrate our theoretical results, showing us that time delay, pulse vaccination, and pulse population input can exert a significant influence on the dynamics of the system which confirms the availability of pulse vaccination strategy for the practical epidemic prevention. Moreover, it is worth pointing out that we obtained an epidemic control strategy for controlling the number of population input

A computational investigation of seasonally forced disease dynamics

In recent years there has been a great increase in work on epidemiological modelling, driven partly by the increase in the availability and power of computers, but also by the desire to improve standards of public and animal health. Through modelling, understanding of the mechanisms of previous epidemics can be gained, and the lessons learnt applied to make predictions about future epidemics, or emerging diseases. The standard SIR model is in some sense quite a simplistic model, and can lack realism. One solution to this problem is to increase the complexity of the model, or to perform full scale simulation—an experiment in silico. This thesis, however, takes a different approach and makes an in depth analysis of one small improvement to the model: the replacement of a constant birth rate with a birth pulse. This more accurately describes the seasonal birth patterns observed in many animal populations. The combination of the nonlinearities of the SIR model and the strong seasonal forcing provided by the birth pulse necessitate the use of numerical methods. The model shows complex multi annual cycles of epidemics and even chaos for shorter infectious periods. The robustness of these results are proven with respect to a wide range or perturbations: in phase space, in the shape and temporal extent of the birth pulse and in the underlying model to which the pulsing is applied. To complement the numerics, analytic methods are used to gain further understanding of the dynamics in particular areas of the chosen parameter space where the numerics can be challenging. Three approximations are presented, one to investigate very small levels of forcing, and two covering short infectious periods.EThOS - Electronic Theses Online ServiceEngineering and Physical Sciences Research Council (EPSRC)GBUnited Kingdo

On the Existence of Equilibrium Points, Boundedness, Oscillating Behavior and Positivity of a SVEIRS Epidemic Model under Constant and Impulsive Vaccination

This paper discusses the disease-free and endemic equilibrium points of a SVEIRS propagation disease model which potentially involves a regular constant vaccination. The positivity of such a model is also discussed as well as the boundedness of the total and partial populations. The model takes also into consideration the natural population growing and the mortality associated to the disease as well as the lost of immunity of newborns. It is assumed that there are two finite delays affecting the susceptible, recovered, exposed, and infected population dynamics. Some extensions are given for the case when impulsive nonconstant vaccination is incorporated at, in general, an aperiodic sequence of time instants. Such an impulsive vaccination consists of a culling or a partial removal action on the susceptible population which is transferred to the vaccinated one. The oscillatory behavior under impulsive vaccination, performed in general, at nonperiodic time intervals, is also discussed

Feedback linearization-based vaccination control strategies for true-mass action type SEIR epidemic models

This paper presents a feedback linearization-based control strategy for a SEIR (susceptible plus infected plus infectious plus removed populations) propagation disease model. The model&nbsp;takes into account the total population amounts as a refrain for the illness transmission since&nbsp;its increase makes more difficult contacts among susceptible and infected. The control objective&nbsp;is novel in the sense that the asymptotically tracking of the removed-by-immunity population&nbsp;to the total population while achieving simultaneously the remaining population (i.e. susceptible&nbsp;plus infected plus infectious) to asymptotically converge to zero. The vaccination policy is firstly&nbsp;designed on the above proposed tracking objective. Then, it is proven that identical vaccination&nbsp;rules might be found based on a general feedback linearization technique. Such a formal technique&nbsp;is very useful in control theory which provides a general method to generate families of vaccination&nbsp;policies with sound technical background which include those proposed in the former sections&nbsp;of the paper. The output zero dynamics of the normal canonical form in the theoretical feedback&nbsp;linearization analysis is identified with that of the removed-by-immunity population. The various&nbsp;proposed vaccination feedback rules involved one of more of the partial populations and there is&nbsp;a certain flexibility in their designs since some control parameters being multiplicative coefficients&nbsp;of the various populations may be zeroed. The basic properties of stability and positivity of the&nbsp;solutions are investigated in a joint way. The equilibrium points and their stability properties as&nbsp;well as the positivity of the solutions are also investigated

Epidemic Models with Pulse Vaccination and Time Delay

In this thesis we discuss deterministic compartmental epidemic models. We study the asymp- totic stability of the disease-free solution of models with pulse vaccination campaigns. The main contributions of this thesis are to extend the literature of pulse vaccination models with delay. We take results for ordinary differential equation models and extend them to models with delay differential equations. Model generalizations include the use of a general incidence term as an upper bound for the actual incidence, and the use of switch parameters to approximate time-varying parameters. In particular, we look at contact rate parameters which are piecewise constant or time-varying. We extend literature results for non-delay general incidence models to find uniform asymptotic stability of the disease-free solution which helps us to add delay. We find an upper bound for the susceptible population under pulse vaccination and use this bound to tighten results for eradication thresholds: that is, we use this upper bound to find sufficient conditions for the uniform asymptotic stability of the disease-free solution of delayed pulse vaccination models. We extend literature results for constant contact rate bilinear incidence delay models to models with periodic time-varying contact rate, and determine conditions under which the disease-free solution is uniformly asymptotically stable for small delay. We also find conditions for disease permanence in the corresponding non-delay, time-varying-parameter pulse vaccination model. For piecewise- constant contact rate bilinear incidence models we again find thresholds which guarantee uniform asymptotic stability under small delay. We additionally discuss the effects of time-varying total population on our results, through a change of variables to population fractions. The total population is commonly held constant in the literature, for analytical simplicity, so we survey the methods for time-varying total population and the effects of such variation on the pulse vaccination schemes. We retain thresholds for eradication by considering the compartment populations as fractions of the total, instead of population numbers. The result is also applied to constant-population delay systems. When changing from standard incidence to bilinear incidence in delay systems, we discuss a way to estimate the effect of time-varying N. We support our theory with simulation results