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On the dynamical complexity of a seasonally forced discrete SIR epidemic model with a constant vaccination strategy
Publication . Rashidinia, Jalil; Sajjadian, Mehri; Duarte, Jorge; Januário, Cristina; Martins, Nuno
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 iteratedmaps 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.
Optimal homotopy analysis of a chaotic HIV-1 model incorporating AIDS-related cancer cells
Publication . Duarte, Jorge; Januário, Cristina; Martins, Nuno; Ramos, Carlos; Rodrigues, Carla; Sardanyés, Josep
The studies of nonlinear models in epidemiology have generated a deep interest in gaining insight into the mechanisms that underlie AIDS-related cancers, providing us with a better understanding of cancer immunity and viral oncogenesis. In this article, we analyze an HIV-1 model incorporating the relations between three dynamical variables: cancer cells, healthy CD4 + T lymphocytes, and infected CD4 + T lymphocytes. Recent theoretical investigations indicate that these cells interactions lead to different dynamical outcomes, for instance to periodic or chaotic behavior. Firstly, we analytically prove the boundedness of the trajectories in the system’s attractor. The complexity of the coupling between the dynamical variables is quantified using observability indices. Our calculations reveal that the highest observable variable is the population of cancer cells, thus indicating that these cells could be monitored in future experiments in order to obtain time series for attractor’s reconstruction. We identify different dynamical behaviors of the system varying two biologically meaningful parameters: r 1, representing the uncontrolled proliferation rate of cancer cells, and k 1, denoting the immune system’s killing rate of cancer cells. The maximum Lyapunov exponent is computed to identify the chaotic regimes. Considering very recent developments in the literature related to the homotopy analysis method (HAM), we calculate the explicit series solutions of the cancer model and focus our analysis on the dynamical variable with the highest observability index. An optimal homotopy analysis approach is used to improve the computational efficiency of HAM by means of appropriate values for the convergence control parameter, which greatly accelerate the convergence of the series solution. The approximated analytical solutions are used to compute density plots, which allow us to discuss additional dynamical features of the model.
Chaos analysis and explicit series solutions to the seasonally forced SIR epidemic model
Publication . Duarte, Jorge; Januário, Cristina; Martins, Nuno; Rogovchenko, Svitlana; Rogovchenko, Yuriy
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.
Controlling infectious diseases: the decisive phase effect on a seasonal vaccination strategy
Publication . Duarte, Jorge; Januário, Cristina; Martins, Nuno; Seoane, Jesús M.; SANJUAN, MIGUEL A. F.
The study of epidemiological systems has generated deep interest in exploring the dynamical complexity of common infectious diseases driven by seasonally varying contact rates. Mathematical modeling and field observations have shown that, under seasonal variation, the incidence rates of some endemic infectious diseases fluctuate dramatically and the dynamics is often characterized by chaotic oscillations in the absence of specific vaccination programs. In fact, the existence of chaotic behavior has been precisely stated in the literature as a noticeable feature in the dynamics of the classical Susceptible-Infected-Recovered (SIR) seasonally forced epidemic model. However, in the context of epidemiology, chaos is often regarded as an undesirable phenomenon associated with the unpredictability of infectious diseases. As a consequence, the problem of converting chaotic motions into regular motions becomes particularly relevant. In this article, we consider the so-called phase control method applied to the seasonally forced SIR epidemic model to suppress chaos. Interestingly, this method of controlling chaos has a clear meaning as a weak perturbation on a seasonal vaccination strategy. Numerical simulations show that the phase difference between the two periodic forces - contact rate and vaccination - plays a very important role in controlling chaos.
The role of noise in the tumor dynamics under chemotherapy treatment
Publication . Bashkirtseva, Irina; Ryashko, Lev; Duarte, Jorge; Seoane, Jesús M.; SANJUAN, MIGUEL A. F.
Dynamical systems modeling tumor growth have been investigated to analyze the dynamics between tumor and healthy cells. Recent theoretical studies indicate that these interactions may lead to different dynamical outcomes under the effect of particular cancer therapies. In the present paper, we derive a system of nonlinear differential equations, in order to investigate solid tumors in vivo, taking into account the impact of chemotherapy on both tumor and healthy cells. We start by studying our model only in terms of deterministic dynamics under the variation of a drug concentration parameter. Later, with the introduction of noise, a stochastic model is used to analyze the impact of the unavoidable random fluctuations. As a result, new insights into noise-induced transitions are provided and illustrated in detail using techniques from dynamical systems and from the theory of stochastic processes.
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Funding agency
Fundação para a Ciência e a Tecnologia
Funding programme
5876
Funding Award Number
UID/MAT/04459/2013