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Abstract

Background Social distancing has led to a “flattening of the curve” in many states across the U.S. This is part of a novel, massive, global social experiment which has served to mitigate the COVID-19 pandemic in the absence of a vaccine or effective anti-viral drugs. Hence it is important to be able to forecast hospitalizations reasonably accurately.

Methods We propose on phenomenological grounds a random walk/generalized diffusion equation which incorporates the effect of social distancing to describe the temporal evolution of the probability of having a given number of hospitalizations. The probability density function is log-normal in the number of hospitalizations, which is useful in describing pandemics where the number of hospitalizations is very high.

Findings We used this insight and data to make forecasts for states using Monte Carlo methods. Back testing validates our approach, which yields good results about a week into the future. States are beginning to reopen at the time of submission of this paper and our forecasts indicate possible precursors of increased hospitalizations. However, the trends we forecast for hospitalizations as well as infections thus far show moderate growth.

Additionally we studied the reproducibility Ro in New York (Italian strain) and California (Wuhan strain). We find that even if there is a difference in the transmission of the two strains, social distancing has been able to control the progression of COVID 19.

Abstract

The wiring diagram of the mouse brain has recently been mapped at a mesoscopic scale in the Allen Mouse Brain Connectivity Atlas. Axonal projections from brain regions were traced using green fluoresent proteins. The resulting data were registered to a common three-dimensional reference space. They yielded a matrix of connection strengths between 213 brain regions. Global features such as closed loops formed by connections of similar intensity can be inferred using tools from persistent homology. We map the wiring diagram of the mouse brain to a simplicial complex (filtered by connection strengths). We work out generators of the first homology group. Some regions, including nucleus accumbens, are connected to the entire brain by loops, whereas no region has non-zero connection strength to all brain regions. Thousands of loops go through the isocortex, the striatum and the thalamus. On the other hand, medulla is the only major brain compartment that contains more than 100 loops.

Abstract

We study the h-discrete and h-discrete fractional representation of a pharmacokinetics-pharmacodynamics (PK-PD) model describing tumor growth and anticancer effects in continuous time considering a time scale h𝕅0, where h > 0. Since the measurements of the drug concentration in plasma were taken hourly, we consider h = 1/24 and obtain the model in discrete time (i.e. hourly). We then continue with fractionalizing the h-discrete nabla operator in the h-discrete model to obtain the model as a system of nabla h-fractional difference equations. In order to solve the fractional h-discrete system analytically we state and prove some theorems in the theory of discrete fractional calculus. After estimating and getting confidence intervals of the model parameters, we compare residual squared sum values of the models in one table. Our study shows that the new introduced models provide fitting as good as the existing models in continuous time.