Unpredictability in seasonal infectious diseases spread (original) (raw)
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Seasonality and period-doubling bifurcations in an epidemic model
Journal of Theoretical Biology, 1984
The annual incidence rates of some endemic infectious diseases are steady while others fluctuate dramatically, often in a regular cycle. In order to investigate the role of seasonality in driving cycles of recurrent epidemics, we analyze numerically the susceptible/exposed/infective/recovered (SEIR) epidemic model with seasonal transmission. We show that smallamplitude periodic solutions exhibit a sequence of period-doubling bifurcations as the amplitude of seasonal variation increases, predicting a transition to chaos of the kind studied in other biological contexts. The epidemiological implication is that the seasonal mechanism generating biennial epidemics may not be able to account for small-amplitude recurrent epidemics of arbitrary periodicity.
Seasonally Forced SIR Systems Applied to Respiratory Infectious Diseases, Bifurcations, and Chaos
Computational and Mathematical Methods
Summary. We investigate models to describe respiratory diseases with fast mutating virus pathogens such that after some years the aquired resistance is lost and hosts can be infected with new variants of the pathogen. Such models were initially suggested for respiartory diseases like influenza, showing complex dynamics in reasonable parameter regions when comparing to historic empirical influenza like illness data, e.g., from Ille de France. The seasonal forcing typical for respiratory diseases gives rise to the different rich dynamical scenarios with even small parameter changes. Especially the seasonality of the infection leads for small values already to period doubling bifurcations into chaos, besides additional coexisting attractors. Such models could in the future also play a role in understanding the presently experienced COVID-19 pandemic, under emerging new variants and with only limited vaccine efficacies against newly upcoming variants. From first period doubling bifurcat...
Effects of quasiperiodic forcing in epidemic models
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2016
We study changes in the bifurcations of seasonally driven compartmental epidemic models, where the transmission rate is modulated temporally. In the presence of periodic modulation of the transmission rate, the dynamics varies from periodic to chaotic. The route to chaos is typically through period doubling bifurcation. There are coexisting attractors for some sets of parameters. However in the presence of quasiperiodic modulation, tori are created in place of periodic orbits and chaos appears via finite torus doublings. Strange nonchaotic attractors (SNAs) are created at the boundary of chaotic and torus dynamics. Multistability is found to be reduced as a function of quasiperiodic modulation strength. It is argued that occurrence of SNAs gives an opportunity of asymptotic predictability of epidemic growth even when the underlying dynamics is strange.
Bifurcation and Chaos in S-I-S Epidemic Model
Differential Equations and Dynamical Systems, 2009
We present a Susceptible-Infective-Susceptible (S-I-S) model with two distinct discrete time delays representing a period of temporary immunity of newborns and a disease incubation period with randomly fluctuating environment. The stability of the equilibria is robustly investigated for the case with and without delay. Conditions for supercritical and subcritical Hopf bifurcation are derived. Comprehensive numerical simulations show that adding delay to an epidemic model could change the asymptotic stability of the system, altering the location of (stable or unstable) endemic equilibrium, or even leading to chaotic behavior. Further, simulation results illustrate that, in some cases where the disease becomes endemic in the model system without delay, addition of delays for temporary immunity and incubation period facilitates smaller final infective population sizes, even if endemicity is still maintained. Effects of randomness of the environment in terms of white noise are thoroughly investigated jointly with delay. The results demonstrate that there are no significant differences in dynamical behaviour of the system when considering delay solely or jointly with stochasticity.
Effects of complexity and seasonality on backward bifurcation in vector-host models
Royal Society open science, 2018
We study implications of complexity and seasonality in vector-host epidemiological models exhibiting backward bifurcation. Vector-host diseases represent complex infection systems that can vary in the transmission processes and population stages involved in disease progression. Seasonal fluctuations in external forcing factors can also interact in a complex way with internal host factors to govern the transmission dynamics. In backward bifurcation, the insufficiency of < 1 for predicting the stability of the disease-free equilibrium (DFE) state arises due to existence of bistability (coexisting DFE and endemic equilibria) for a range of values below one. Here we report that this region of bistability decreases with increasing complexity of vector-borne disease transmission as well as with increasing seasonality strength. The decreases in the bistability region are accompanied by a reduced force of infection acting on primary hosts. As a consequence, we show counterintuitively th...
Chaos analysis and explicit series solutions to the seasonally forced SIR epidemic model
Journal of Mathematical Biology, 2019
Despite numerous studies of epidemiological systems, the role of seasonality in the recurrent epidemics is not entirely understood. During certain periods of the year incidence rates of a number of endemic infectious diseases may fluctuate dramatically. This influences the dynamics of mathematical models describing the spread of infection and often leads to chaotic oscillations. In this paper, we are concerned with a generalization of a classical Susceptible-Infected-Recovered epidemic model which accounts for seasonal effects. Combining numerical and analytic techniques, we gain new insights into the complex dynamics of a recurrent disease influenced by the seasonality. Computation of the Lyapunov spectrum allows us to identify different chaotic regimes, determine the fractal dimension and estimate the predictability of the appearance of attractors in the system. Applying the homotopy analysis method, we obtain series solutions to the original nonautonomous SIR model with a high level of accuracy and use these approximations to analyze the dynamics of the system. The efficiency of the method is guaranteed by the optimal choice of an auxiliary control parameter which ensures the rapid convergence of the series to the exact solution of the forced SIR epidemic model.
Bifurcation and chaos in an epidemic model with nonlinear incidence rates
Applied Mathematics and Computation, 2010
This paper investigates a discrete-time epidemic model by qualitative analysis and numerical simulation. It is verified that there are phenomena of the transcritical bifurcation, flip bifurcation, Hopf bifurcation types and chaos. Also the largest Lyapunov exponents are numerically computed to confirm further the complexity of these dynamic behaviors. The obtained results show that discrete epidemic model can have rich dynamical behavior.
Journal of Mathematical Biology, 1985
A seasonally forced nonlinear SEIR epidemic model is used to simulate small and large amplitude periodic outbreaks. The model is shown to exhibit bistable behavior for a fixed set of parameters. Basins of attraction for each recurrent outbreak are computed, and it is shown that the basins of two coexisting stable outbreaks are intertwined in a complicated manner. The effect of such a basin structure is shown to result in an obstruction in predicting asymptotically the type of outbreak given an uncertainty in the initial population of susceptibles and infectives.
Dynamical analysis and chaos control of a discrete SIS epidemic model
Advances in Difference Equations, 2014
The dynamical behaviors of a discrete-time SIS epidemic model are investigated in this paper. The result indicates that the model undergoes a flip bifurcation and a Hopf bifurcation, as found by using the center manifold theorem and bifurcation theory. Numerical simulations not only illustrate our results, but they also exhibit the complex dynamical behaviors, such as the period-doubling bifurcation in period-2, -4, -8, quasi-periodic orbits and the chaotic sets. Specifically, when the parameters A, d 1 , d 2 , r, λ are fixed at some values and the bifurcation parameter h changes with different values, there exist local stability, Hopf bifurcation, 3-periodic orbits, 7-periodic orbits, period-doubling bifurcation and chaotic sets. These results reveal far richer dynamical behaviors of the discrete epidemic model compared with the continuous epidemic models although the discrete epidemic model is simple. Finally, the feedback control method is used to stabilize chaotic orbits at an unstable endemic equilibrium.
Evidence of chaos in eco-epidemic models
Mathematical Biosciences and Engineering, 2009
We study an eco-epidemic model with two trophic levels in which the dynamics is determined by predator-prey interactions as well as the vulnerability of the predator to a disease. Using the concept of generalized models we show that for certain classes of eco-epidemic models quasiperiodic and chaotic dynamics is generic and likely to occur. This result is based on the existence of bifurcations of higher codimension such as double Hopf bifurcations. We illustrate the emergence of chaotic behavior with one example system.