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The SIR epidemiological equations model new affected and removed cases as roughly proportional to the current number of infected cases. The present report adopts an alternative that has been considered in the literature, in which the number of new affected cases is proportional to the α ≤ 1 power of the number of infected cases. After arguing that α = 1 models exponential growth while α < 1 models polynomial growth, a simple method for parameter estimation in differential equations subject to noise, the random-time transformation RTT of Bassan, Meilijson, Marcus and Talpaz 1997, will be reviewed and applied in an attempt to settle the question as to the nature of Covid19.
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We show that semiparametric profile likelihoods, where the nuisance parameter has been profiled out, behave like ordinary likelihoods in that they have a quadratic expansion. In this expansion the score function and the Fisher information are replaced by-the efficient score function and efficient Fisher information. The expansion may be used, among others, to prove the asymptotic normality of the maximum likelihood estimator, to derive the asymptotic chi-squared distribution of the log-likelihood ratio statistic, and to prove the consistency of the observed information as an estimator of the inverse of the asymptotic variance.
Differential equations with measurements subject to errors are usually handled by Least Squares methods or by Likelihood methods based on diffusion-type stochastic modifications of the differential equation. We study the performance of likelihood methods based on substituting a Gaussian random time transformation as argument in the solution of the original deterministic differential equation. This method may be applied to the simultaneous estimation of parameters describing a number of differential equations, based on data with dependent measurement errors. The model is fitted to disease progress curves derived from a real data set consisting of disease assessments of melon plants infected by Zucchini Yellow Mosaic Virus (ZYMV). KeywordsOrnstein-Uhlenbeck process-plant disease epidemiology
The investigation of a distance-based regression model, using a one-dimensional set of equally spaced points as regressor values and √|x − y| as a distance function, leads to the study of a family of matrices which is closely related to a discrete analog of the Brownian-bridge stochastic process. We describe its eigenstructure and several properties, recovering in particular well-known results on tridiagonal Toeplitz matrices and related topics.
Two key linked questions in Population dynamics are the relative importance of noise vs. density-dependent nonlinearities and the limits on temporal predictability of population abundance. We propose that childhood microparasitic infections, notably measles. provide an unusually suitable empirical and theoretical test bed for addressing these issues. We base our analysis on a new mechanistic time series model for measles, the TSIR model. which captures the mechanistic essence of epidemic dynamics. The model, and parameter estimates based on short-term fits to prevaccination measles time series for 60 towns and cities in England and Wales, is introduced in a companion paper. Here. we explore how well the model predicts the long-term dynamics of measles and the balance between noise and determinism, as a function of population size. The TSIR model captures the basic dynamical features of the long-term pattern of measles epidemics in large cities remarkably well (based on time and frequency domain analyses). In particular, the model illustrates the impact of secular increases in birth rates, which cause a transition from biennial to annual dynamics. The model also captures the observed increase in epidemic irregularity with decreasing population size and the onset of local extinction below a critical community size. Decreased host population size is shown to be associated with an increased impact of demographic stochasticity. The interaction between nonlinearity and noise is explored using local Lyapunov exponents (LLE). These testify to the high level of stability of the biennial attractor in large cities. Irregularities are due to the limit cycle evolving with changing human birth rates and not due to complex dynamics. The geometry of the dynamics (sign and magnitude of the LLEs across phase space) is similar in the cities and the smaller urban areas. The qualitative difference in dynamics between small and large host communities is that demographic and extinction-recolonization stochasticities are much more influential in the former. The regional dynamics can therefore only be understood in terms of a core-satellite metapopulation structure for this host-enemy system. We also make a preliminary exploration of the model's ability to predict the dynamic consequences of measles vaccination.
The effect of drift change on Skorohod embedded distribution with applications in Finance
  • N Alon
Alon N. (2019), The effect of drift change on Skorohod embedded distribution with applications in Finance. Master's Thesis, Tel Aviv University