arXiv subjects

Sana Jahedi

Publications and source records attributed to Sana Jahedi.

3 recordsLinked to original sources

arXiv2024

Ambiguity in the use of SIR models to fit epidemic incidence data

When fitting a multi-parameter model to a data set, computer algorithms may suggest that a range of parameters provide equally reasonable fits, making the parameter estimation difficult. Here, we prove this fact for an SIR model. We say a set of parameter values is a good fit to outbreak data if the solution has the data's three most significant characteristics: the standard deviation, the mean time, and the total number of cases. In our model, in addition to the "basic reproduction number" $R_0$, three other parameters need to be estimated to fit a solution to outbreak data. We will show that those parameters can be chosen so that each gives a linear transformation of a solution's incidence data. As a result, we show that for every choice of $R_0>1$, there is a good fit for each outbreak. We also illustrate our results by providing the least square best fits of the New York City and London data sets of the Omicron variant of COVID-19. Furthermore, we show how versions of the SIR model with $N$ compartments have far more good fits- - indeed a high dimensional set of good fits -- for each target -- showing that more complicated models may have an even greater problem in overparametrizing outbreak characteristics.

q-bio.PE↗

arXiv2023

Structured Systems of Nonlinear Equations

In a "structured system" of equations, each equation depends on a specified subset of the variables. In this article, we explore properties common to "almost every" system with a fixed structure and how the properties can be read from the corresponding connection graph. A solution $p$ of a system $F(p)=c$ is called robust if it persists despite small changes in $F$. We establish methods for determining robustness that depends on the structure, as expressed in the properties of the corresponding directed graph of the structured system. The keys to understanding linear and nonlinear structured systems are subsets of variables that we call forward and backward bottlenecks. In particular, when robustness fails in a structured system, it is due to the existence of a unique "backward bottleneck", that we call a "minimax bottleneck". We present a numerical method for locating the minimax bottleneck. We show how to remove it by adding edges to the graph.

math.CA↗

arXiv2022

Robustness of solutions of almost every system of equations

In mathematical modeling, it is common to have an equation $F(p)=c$ where the exact form of $F$ is not known. This article shows that there are large classes of $F$ where almost all $F$ share the same properties. The classes we investigate are vector spaces $\mathcal{F}$ of $C^1$ functions $F:\mathbb{R}^N \to \mathbb{R}^M$ that satisfy the following condition: $\mathcal{F}$ has ``almost constant rank'' (ACR) if there is a constant integer $ρ(\mathcal{F}) \geq 0$ such that rank$(DF(p))=ρ(\mathcal{F})$ for ``almost every'' $F\in \mathcal{F}$ and almost every $p\in\mathbb{R}^N$. If the vector space $\mathcal{F}$ is finite-dimensional, then ``almost every'' is with respect to Lebesgue measure on $\mathcal{F}$, and otherwise, it means almost every in the sense of prevalence, as described herein. Most function spaces commonly used for modeling purposes are ACR. In particular, we show that if all of the functions in $\mathcal{F}$ are linear or polynomial or real analytic, or if $\mathcal{F}$ is the set of all functions in a ``structured system'', then $\mathcal{F}$ is ACR. For each $F$ and $p$, the solution set of $p \in \mathbb{R}^N$ is SolSet$(p):= \{x: F(x)=F(p)\}.$ A solution set of $F(p)=c$ is called robust if it persists despite small changes in $F$ and $c$. The following two global results are proved for almost every $F$ in an ACR vector space $\mathcal{F}$: (1) Either the solution set SolSet$(p)$ is robust for almost every $p\in \mathbb{R}^N$, or none of the solution sets are robust. (2) The solution set SolSet$(p)$ is a $C^\infty$-manifold of dimension $d = N-ρ(\mathcal{F})$. In particular, $d$ is the same for almost every $F \in \mathcal{F}$.

math.CA↗