On the Dynamical Complexity of a Seasonally Forced Discrete SIR Epidemic Model with a Constant Vaccination Strategy

Abstract

In this article, we consider the discretized classical Susceptible-Infected-Recovered (SIR) forced epidemic model to investigate the consequences of the introduction of different transmission rates and the effect of a constant vaccination strategy, providing new numerical and topological insights into the complex dynamics of recurrent diseases. Starting with a constant contact (or transmission) rate, the computation of the spectrum of Lyapunov exponents allows us to identify different chaotic regimes. Studying the evolution of the dynamical variables, a family of unimodal-type iterated maps with a striking biological meaning is detected among those dynamical regimes of the densities of the susceptibles. Using the theory of symbolic dynamics, these iterated maps are characterized based on the computation of an important numerical invariant, the topological entropy. The introduction of a degree (or amplitude) of seasonality, ε, is responsible for inducing complexity into the population dynamics. The resulting dynamical behaviors are studied using some of the previous tools for particular values of the strength of the seasonality forcing, ε. Finally, we carry out a study of the discrete SIR epidemic model under a planned constant vaccination strategy. We examine what effect this vaccination regime will have on the periodic and chaotic dynamics originated by seasonally forced epidemics.

Individualism or Collectivism: A Reinforcement Learning Mechanism for Vaccination Decisions

“…where N is the total number of individuals in the community and n v is the number of vaccinated individuals in the community. In the context of epidemic modeling, R 0 is usually defined as the so-called basic reproductive rate [42]. If we consider relative cost c rather than separately considering vaccine costs C V and infection costs C I , we can modify Equation (4) as follows,…”

Analytical Modeling of the Temporal Evolution of Epidemics Outbreaks Accounting for Vaccinations

“…There are several variants [2][3][4][5][6][7][8][9][10] of the SIR model, including stochastic variants [11][12][13][14][15][16][17][18][19][20][21][22][23][24] and there is a large amount of related work in the context of the SIR model in the presence of vaccination. Many works deal with vaccination strategies [13,[25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43], transmission among an interconnected group or population [44] and vaccination behavior by coupling the epidemic spreading with human decisions [45], or policies [46,47] using the SIR model with limited resources [48]. One of the classical v...…”

Fractional-Order Discrete-Time SIR Epidemic Model with Vaccination: Chaos and Complexity

“…To investigate the dynamic of the commensurate fractional-order discrete-time SIR epidemic model with vaccination given in (11) regarding the fractional order γ = γ 1 = γ 2 , we set the parameters N = 100, β = 0.8, σ = 0.1, p = 0.005, and initial conditions (S(0), I(0)) = (70, 30). Figure 3 displays the phase space of model ( 11) for γ = 1.…”

“…The effect of vaccination on the dynamics of an epidemic model has been studied by adding a component for the vaccinated individuals to the epidemic model and obtaining the epidemic model with vaccination. For example, in [11] the authors studied the dynamical behavior of a discrete SIR epidemic model with a constant vaccination strategy. The stability and bifurcations of a discrete susceptible-infected-susceptible (SIS) epidemic model with vaccination were investigated in [12], while Xiang et al in [13] developed a discrete SIRS model that included vaccination and examined its dynamical behavior.…”