410 research outputs found
Equation-Free Multiscale Computational Analysis of Individual-Based Epidemic Dynamics on Networks
The surveillance, analysis and ultimately the efficient long-term prediction
and control of epidemic dynamics appear to be one of the major challenges
nowadays. Detailed atomistic mathematical models play an important role towards
this aim. In this work it is shown how one can exploit the Equation Free
approach and optimization methods such as Simulated Annealing to bridge
detailed individual-based epidemic simulation with coarse-grained,
systems-level, analysis. The methodology provides a systematic approach for
analyzing the parametric behavior of complex/ multi-scale epidemic simulators
much more efficiently than simply simulating forward in time. It is shown how
steady state and (if required) time-dependent computations, stability
computations, as well as continuation and numerical bifurcation analysis can be
performed in a straightforward manner. The approach is illustrated through a
simple individual-based epidemic model deploying on a random regular connected
graph. Using the individual-based microscopic simulator as a black box
coarse-grained timestepper and with the aid of Simulated Annealing I compute
the coarse-grained equilibrium bifurcation diagram and analyze the stability of
the stationary states sidestepping the necessity of obtaining explicit closures
at the macroscopic level under a pairwise representation perspective
Stability and bifurcations in an epidemic model with varying immunity period
An epidemic model with distributed time delay is derived to describe the
dynamics of infectious diseases with varying immunity. It is shown that
solutions are always positive, and the model has at most two steady states:
disease-free and endemic. It is proved that the disease-free equilibrium is
locally and globally asymptotically stable. When an endemic equilibrium exists,
it is possible to analytically prove its local and global stability using
Lyapunov functionals. Bifurcation analysis is performed using DDE-BIFTOOL and
traceDDE to investigate different dynamical regimes in the model using
numerical continuation for different values of system parameters and different
integral kernels.Comment: 16 pages, 5 figure
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Travelling waves for an epidemic model with non-smooth treatment rates
This is the post-print version of the final published paper that is available from the link below. Copyright @ 2010 IOP Publishing Ltd and SISSA.We consider a susceptible–infected–removed (SIR) epidemic model with two types of nonlinear treatment rates: (i) piecewise linear treatment rate with saturation effect, (ii) piecewise constant treatment rate with a jump (Heaviside function). For case (i), we compute travelling front solutions whose profiles are heteroclinic orbits which connect either the disease-free state to an infective state or two endemic states with each other. For case (ii), it is shown that the profile has the following properties: the number of susceptibles is monotonically increasing and the number of infectives approaches zero at infinity, while their product converges to a constant. Numerical simulations are performed for all these cases. Abnormal behaviour like travelling waves with non-monotonic profile or oscillations is observed
Existence of periodic solutions for the periodically forced SIR model
We prove that the seasonally-forced SIR model with a T-periodic forcing has a
periodic solution with period T whenever the basic reproductive number R0>1.
The proof uses the Leray-Schauder degree theory. We also describe some
numerical results in which we compute the T-periodic solution, where in order
to obtain the T-periodic solution when the behavior of the system is
subharmonic or chaotic, we use a Galerkin scheme
Unpredictability in seasonal infectious diseases spread
In this work, we study the unpredictability of seasonal infectious diseases
considering a SEIRS model with seasonal forcing. To investigate the dynamical
behaviour, we compute bifurcation diagrams type hysteresis and their respective
Lyapunov exponents. Our results from bifurcations and the largest Lyapunov
exponent show bistable dynamics for all the parameters of the model. Choosing
the inverse of latent period as control parameter, over 70% of the interval
comprises the coexistence of periodic and chaotic attractors, bistable
dynamics. Despite the competition between these attractors, the chaotic ones
are preferred. The bistability occurs in two wide regions. One of these regions
is limited by periodic attractors, while periodic and chaotic attractors bound
the other. As the boundary of the second bistable region is composed of
periodic and chaotic attractors, it is possible to interpret these critical
points as tipping points. In other words, depending on the latent period, a
periodic attractor (predictability) can evolve to a chaotic attractor
(unpredictability). Therefore, we show that unpredictability is associated with
bistable dynamics preferably chaotic, and, furthermore, there is a tipping
point associated with unpredictable dynamics
Dynamic analysis of a fractional-order SIRS model with time delay
Mathematical modeling plays a vital role in the epidemiology of infectious diseases. Policy makers can provide the effective interventions by the relevant results of the epidemic models. In this paper, we build a fractional-order SIRS epidemic model with time delay and logistic growth, and we discuss the dynamical behavior of the model, such as the local stability of the equilibria and the existence of Hopf bifurcation around the endemic equilibrium. We present the numerical simulations to verify the theoretical analysis
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