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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 susceptibleāinfectedāremoved (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
A simple SIR model with a large set of asymptomatic infectives
There is increasing evidence that one of the most difficult problems in
trying to control the ongoing COVID-19 epidemic is the presence of a large
cohort of asymptomatic infectives. We develop a SIR-type model taking into
account the presence of asymptomatic, or however undetected, infective, and the
substantially long time these spend being infective and not isolated. We
discuss how a SIR-based prediction of the epidemic course based on early data
but not taking into account the presence of a large set of asymptomatic
infectives would give wrong estimate of very relevant quantities such as the
need of hospital beds, the time to the epidemic peak, and the number of people
which are left untouched by the first wave and thus in danger in case of a
second epidemic wave. In the second part of the note, we apply our model to the
COVID-19 epidemics in Italy. We obtain a good agreement with epidemiological
data; according to the best fit of epidemiological data in terms of this model,
only 10\% of infectives in Italy is symptomatic.Comment: V4 (hopefully final) contains analysis of data up to May 15, 202
Pulse vaccination in the periodic infection rate SIR epidemic model
A pulse vaccination SIR model with periodic infection rate have
been proposed and studied. The basic reproductive number 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 and is unstable if
. Standard bifurcation theory have been used to show the existence of
the positive periodic solution for the case of . 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
Fitting stochastic epidemic models to gene genealogies using linear noise approximation
Phylodynamics is a set of population genetics tools that aim at
reconstructing demographic history of a population based on molecular sequences
of individuals sampled from the population of interest. One important task in
phylodynamics is to estimate changes in (effective) population size. When
applied to infectious disease sequences such estimation of population size
trajectories can provide information about changes in the number of infections.
To model changes in the number of infected individuals, current phylodynamic
methods use non-parametric approaches, parametric approaches, and stochastic
modeling in conjunction with likelihood-free Bayesian methods. The first class
of methods yields results that are hard-to-interpret epidemiologically. The
second class of methods provides estimates of important epidemiological
parameters, such as infection and removal/recovery rates, but ignores variation
in the dynamics of infectious disease spread. The third class of methods is the
most advantageous statistically, but relies on computationally intensive
particle filtering techniques that limits its applications. We propose a
Bayesian model that combines phylodynamic inference and stochastic epidemic
models, and achieves computational tractability by using a linear noise
approximation (LNA) --- a technique that allows us to approximate probability
densities of stochastic epidemic model trajectories. LNA opens the door for
using modern Markov chain Monte Carlo tools to approximate the joint posterior
distribution of the disease transmission parameters and of high dimensional
vectors describing unobserved changes in the stochastic epidemic model
compartment sizes (e.g., numbers of infectious and susceptible individuals). We
apply our estimation technique to Ebola genealogies estimated using viral
genetic data from the 2014 epidemic in Sierra Leone and Liberia.Comment: 43 pages, 6 figures in the main tex
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