Quantitative Biology

The ongoing COVID-19 pandemic is challenging every part of society. From a scientific point of view the first major task is to predict the dynamics of the pandemic, allowing governments to allocate proper resources and measures to fight it, as well as gauging the success of these measures by comparison with the predictions in hindsight. The vast majority of pandemic models are based on extensive models with large numbers of fit parameters, leading to individual descriptions for every hot spot on the world. This makes predictions and comparisons cumbersome, if not impossible. We here propose a different approach, by moving away from a description over time, and instead choosing the total number of infected people in an enclosed area as the independent variable. Analyzing a few hot spots data, we derive an empirical formula for the dynamics, dependent only on three variables. The final number of infections is strictly connected to one fit parameter we call mitigation factor, which in turn is mostly dependent only on the enclosed population. Despite its simpleness, this description applies to every of the around 50 countries we have analyzed, allows to separate different waves of the pandemic, provides a figure of merit for the overall usefulness of government measures, and shows when a pandemic is ending. Our model is robust against undetected cases, and allows all nations, in particular those with fewer resources, to reasonably predict the outcome of the pandemic in their country.

An accurate estimation of the Protein Space size, in light of the factors that govern it, is a long-standing problem and of paramount importance in evolutionary biology, since it determines the nature of protein evolvability. A simple analysis will enable us to, firstly, reduce an unrealistic Protein Space size of ~10^130 sequences, for a 100-residues polypeptide chain, to ~10^9 functional proteins and, secondly, estimate a robust average-mutation rate per amino acid (x ~1.23) and infer from it, in light of the protein marginal stability, that only a fraction of the sequence will be available at any one time for a functional protein to evolve. Although this result does not solve the Protein Space vastness problem, frames it in a more rational one.

We investigate the evolutionary rescue of a microbial population in a gradually deteriorating environment, through a combination of analytical calculations and stochastic simulations. We consider a population destined for extinction in the absence of mutants, which can only survive if mutants sufficiently adapted to the new environment arise and fix. We show that mutants that appear later during the environment deterioration have a higher probability to fix. The rescue probability of the population increases with a sigmoidal shape when the product of the carrying capacity and of the mutation probability increases. Furthermore, we find that rescue becomes more likely for smaller population sizes and/or mutation probabilities if the environment degradation is slower, which illustrates the key impact of the rapidity of environment degradation on the fate of a population. We also show that our main conclusions are robust across various types of adaptive mutants, including specialist and generalist ones, as well as mutants modeling antimicrobial resistance evolution. We further express the average time of appearance of the mutants that do rescue the population and the average extinction time of those that do not. Our methods can be applied to other situations with continuously variable fitnesses and population sizes, and our analytical predictions are valid in the weak-to-moderate mutation regime.

In constant parameter compartmental models an early onset of herd immunity is at odds with estimates of R values from early stage growth. This paper utilizes a result from the theory of interest rate modeling, namely a bond pricing formula of Vasicek, and an approach inspired by a foundational result in statistics, de Finetti's Theorem, to show how the modeling discrepancy can be explained. Moreover the difference between predictions of classic constant parameter epidemiological models and those with variation and stochastic evolution can be reduced to simple "convexity" formulas. A novel feature of this approach is that we do not attempt to locate a true model but only a model that is equivalent after permutations. Convexity adjustments can also be used for cross sectional comparisons and we derive easy to use rules of thumb for estimating threshold infection level in one region given knowledge of threshold infection in another.

In this article, we discuss an age-structured SIR model in which disease not only spread through direct person to person contacts for e.g. infection due to surface contamination but it can also spread through indirect contacts. It is evident that age also plays a crucial role in SARS virus infection including COVID-19 infection. We formulate our model as an abstract semilinear Cauchy problem in an appropriate Banach space to show the existence of solution and also show the existence of steady states. It is assumed in this work that the population is in a demographic stationary state and show that there is no disease-free equilibrium point as long as there is a transmission of infection due to the indirect contacts in the environment.

We propose an epidemic model SIPHERD in which three categories of infection carriers Symptomatic, Purely Asymptomatic, and Exposed are considered with different rates of transmission of infection that are taken dependent on the lockdown and social distancing. The rate of detection of the infected carriers is taken dependent on the tests done per day. The model is applied for the COVID outbreak in Germany and South Korea to validate its predictive capabilities and then applied to India and the United States for the prediction of its spread with different lockdown situations and testing in the coming months.

We present a compartmental meta-population model for the spread of Covid-19 in India. Our model simulates populations at a district or state level using an epidemiological model that is appropriate to Covid-19. Different districts are connected by a transportation matrix developed using available census data. We introduce uncertainties in the testing rates into the model that takes into account the disparate responses of the different states to the epidemic and also factors in the state of the public healthcare system. Our model allows us to generate qualitative projections of Covid-19 spread in India, and further allows us to investigate the effects of different proposed interventions. By building in heterogeneity at geographical and infrastructural levels and in local responses, our model aims to capture some of the complexity of epidemiological modeling appropriate to a diverse country such as India.

The global COVID-19 pandemic (SARS-CoV-2 virus) is the defining health crisis of our century. Due to the absence of vaccines and drugs that can help to fight it, the world solution to control the spread has been to consider public social distance measures that avoids the saturation of the health system. In this context, we investigate a Model Predictive Control (MPC) framework to determine the time and duration of social distancing policies. We use Brazilian data in the period from March to May of 2020. The available data regarding the number of infected individuals and deaths suffers from sub-notification due to the absence of mass tests and the relevant presence of the asymptomatic individuals. We estimate variations of the SIR model using an uncertainty-weighted Least-Squares criterion that considers both nominal and inconsistent-data conditions. Moreover, we add to our versions of the SIR model an additional dynamic state variable to mimic the response of the population to the social distancing policies determined by the government that affects the speed of COVID-19 transmission. Our control framework is within a mixed-logical formalism, since the decision variable is forcefully binary (the existence or the absence of social distance policy). A dwell-time constraint is included to avoid harsh shifting between these two states. Finally, we present simulation results to illustrate how such optimal control policy would operate. These results point out that no social distancing should be relaxed before mid August 2020. If relaxations are necessary, they should not be performed before the beginning this date and should be in small periods, no longer than 25 days. This paradigm would proceed roughly until January/2021. The second peak of infections, which has a forecast to the beginning of October, can be reduced if the periods of no-isolation days are shortened.

Recent work has proven the existence of extreme inbreeding in a European ancestry sample taken from the contemporary UK population \cite{nature_01}. This result brings our attention again to a math problem related to inbreeding family trees and diversity. Groups with a finite number of individuals could give a variety of genetic relationships. { In previous works \cite{PhysRevE.92.052132, PhysRevE.90.022125, JARNE20191}, we have addressed the issue of building inbreeding trees for biparental reproduction using Markovian models. Here, we extend these studies by presenting an algorithm to generate and represent inbreeding trees with no overlapping generations. We explicitly assume a two-gender reproductory scheme, and we pay particular attention to the links between nodes. We show that even for a simple case with a relatively small number of nodes in the tree, there are a large number of possible ways to rearrange the links between generations. We present an open-source python code to generate the tree graph, the adjacency matrix, and the histogram of the links for each different tree representation. We show how this mapping reflects the difference between tree realizations, and how valuable information may be extracted upon inspection of these matrices. The algorithm includes a feature to average several tree realizations, obtain the connectivity distribution, and calculate the average and mean value. We used this feature to compare trees with a different number of generations and nodes. The code presented here, available in Git-Hub, may be easily modified to be applied to other areas of interest involving connections between individuals, extend the study to add more characteristics of the different nodes, etc.

This paper considers an explicit numerical scheme for solving the mathematical model of the propagation of Covid-19 epidemic with undetected infectious cases. We analyze the stability and convergence rate of the new approach in L ∞ -norm. The proposed method is less time consuming. Furthermore, the method is stable, at least second-order convergent and can serve as a robust tool for the integration of general systems of ordinary differential equations. A wide set of numerical evidences which consider the case of Cameroon are presented and discussed.