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

Nonlinear pulse vaccination in an SIR epidemic model with resource limitation

Mathematical models can assist in the design and understanding of vaccination strategies when resources are limited. Here we propose and analyse an SIR epidemic modelwith a nonlinear pulse vaccination to examine how a limited vaccine resource affects the transmission and control of infectious diseases, in particular emerging infectious diseases. The threshold condition for the stability of the disease free steady state is given. Latin Hypercube Sampling/Partial Rank Correlation Coefficient uncertainty and sensitivity analysis techniques were employed to determine the key factors which are most significantly related to the threshold value. Comparing this threshold value with that without resource limitation, our results indicate that if resources become limited pulse vaccination should be carried out more frequently than when sufficient resources are available to eradicate an infectious disease. Once the threshold value exceeds a critical level, both susceptible and infected populations can oscillate periodically. Furthermore, when the pulse vaccination period is chosen as a bifurcation parameter, the SIR model with nonlinear pulse vaccination reveals complex dynamics including period doubling, chaotic solutions, and coexistence of multiple attractors. The implications of our findings with respect to disease control are discussed

The dynamics of measles in sub-Saharan Africa.

Although vaccination has almost eliminated measles in parts of the world, the disease remains a major killer in some high birth rate countries of the Sahel. On the basis of measles dynamics for industrialized countries, high birth rate regions should experience regular annual epidemics. Here, however, we show that measles epidemics in Niger are highly episodic, particularly in the capital Niamey. Models demonstrate that this variability arises from powerful seasonality in transmission-generating high amplitude epidemics-within the chaotic domain of deterministic dynamics. In practice, this leads to frequent stochastic fadeouts, interspersed with irregular, large epidemics. A metapopulation model illustrates how increased vaccine coverage, but still below the local elimination threshold, could lead to increasingly variable major outbreaks in highly seasonally forced contexts. Such erratic dynamics emphasize the importance both of control strategies that address build-up of susceptible individuals and efforts to mitigate the impact of large outbreaks when they occur

SIR model with vaccination: bifurcation analysis

There are few adapted SIR models in the literature that combine vaccination and logistic growth. In this article, we study bifurcations of a SIR model where the class of Susceptible individuals grows logistically and has been subject to constant vaccination. We explicitly prove that the endemic equilibrium is a codimension two singularity in the parameter space $(\mathcal{R}_0, p)$, where $\mathcal{R}_0$ is the basic reproduction number and $p$ is the proportion of Susceptible individuals successfully vaccinated at birth. We exhibit explicitly the Hopf, transcritical, Belyakov, heteroclinic and saddle-node bifurcation curves unfolding the singularity. The two parameters $(\mathcal{R}_0, p)$ are written in a useful way to evaluate the proportion of vaccinated individuals necessary to eliminate the disease and to conclude how the vaccination may affect the outcome of the epidemic. We also exhibit the region in the parameter space where the disease persists and we illustrate our main result with numerical simulations, emphasizing the role of the parameters

Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay

Pulse vaccination, the repeated application of vaccine over a defined age range, is gaining prominence as an effective strategy for the elimination of infectious diseases. An SIR epidemic model with pulse vaccination and distributed time delay is proposed in this paper. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact infection-free periodic solution of the impulsive epidemic system and prove that the infection-free periodic solution is globally attractive if the vaccination rate is larger enough. Moreover, we show that the disease is uniformly persistent if the vaccination rate is less than some critical value. The permanence of the model is investigated analytically. Our results indicate that a large pulse vaccination rate is sufficient for the eradication of the disease

Seasonality in epidemic models: a literature review

We provide a review of some key literature results on the influence of seasonality and other time heterogeneities of contact rates, and other parameters, such as vaccination rates, on the spread of infectious diseases. This is a classical topic where highly theoretical methodologies have provided new insight on the seemingly random behavior observed in epidemic time-series. We follow the line of providing a highly personal non-systematic review of this topic, mainly based on the history of mathematical epidemiology and on the impact of reviewed articles. Our aim is to stress some issues of increasing interest, such as the public health implications of the biomathematical literature and the impact of seasonality on epidemic extinction or elimination

Dynamics of vaccination in a time-delayed epidemic model with awareness

This paper investigates the effects of vaccination on the dynamics of infectious disease, which is spreading in a population concurrently with awareness. The model considers contributions to the overall awareness from a global information campaign, direct contacts between unaware and aware individuals, and reported cases of infection. It is assumed that there is some time delay between individuals becoming aware and modifying their behaviour. Vaccination is administered to newborns, as well as to aware individuals, and it is further assumed that vaccine-induced immunity may wane with time. Feasibility and stability of the disease-free and endemic equilibria are studied analytically, and conditions for the Hopf bifurcation of the endemic steady state are found in terms of system parameters and the time delay. Analytical results are supported by numerical continuation of the Hopf bifurcation and numerical simulations of the model to illustrate different types of dynamical behaviour

Scaling Symmetries and Parameter Reduction in Epidemic SI(R)S Models

Symmetry concepts in parametrized dynamical systems may reduce the number of external parameters by a suitable normalization prescription. If, under the action of a symmetry group G , parameter space A becomes a (locally) trivial principal bundle, A ≅ A / G × G , then the normalized dynamics only depends on the quotient A / G . In this way, the dynamics of fractional variables in homogeneous epidemic SI(R)S models, with standard incidence, absence of R-susceptibility and compartment independent birth and death rates, turns out to be isomorphic to (a marginally extended version of) Hethcote’s classic endemic model, first presented in 1973. The paper studies a 10-parameter master model with constant and I-linear vaccination rates, vertical transmission and a vaccination rate for susceptible newborns. As recently shown by the author, all demographic parameters are redundant. After adjusting time scale, the remaining 5-parameter model admits a 3-dimensional abelian scaling symmetry. By normalization we end up with Hethcote’s extended 2-parameter model. Thus, in view of symmetry concepts, reproving theorems on endemic bifurcation and stability in such models becomes needless

Endemic Oscillations for SARS-CoV-2 Omicron -- A SIRS model analysis

The SIRS model with constant vaccination and immunity waning rates is well known to show a transition from a disease-free to an endemic equilibrium as the basic reproduction number $r_0$ is raised above threshold. It is shown that this model maps to Hethcote's classic endemic model originally published in 1973. In this way one obtains unifying formulas for a whole class of models showing endemic bifurcation. In particular, if the vaccination rate is smaller than the recovery rate and $r_- < r_0 < r_+$ for certain upper and lower bounds $r_\pm$, then trajectories spiral into the endemic equilibrium via damped infection waves. Latest data of the SARS-CoV-2 Omicron variant suggest that according to this simplified model continuous vaccination programs will not be capable to escape the oscillating endemic phase. However, in view of the strong damping factors predicted by the model, in reality these oscillations will certainly be overruled by time-dependent contact behaviors.Comment: 19 pages, 9 figure

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