arXiv subjects

Florin Avram

Publications and source records attributed to Florin Avram.

At least 19 recordsLinked to original sources

arXiv2026

A Perron-Frobenius strong threshold theorem for (A, B, P, {\phi}) balanced bilinear models, and the role of left and right Perron eigenvectors in mathematical epidemiology

This paper started as a review of seven results pertaining to a family of bilinear models with rank one NGM introduced by Fall, Iggidr, Sallet and Bonzi, which utilize the explicit eigenvectors of the NGM to compute the unique endemic equilibrium (EE), and Lyapunov functions at both the disease free equilibrium (DFE) and EE, and of results of Shuai and Van den Driessche (2013), which essentially deal with the same "DFE -EE stability exchange" in the non-rank one case, when the eigenvectors are not explicit. Recently, these results were complemented by Earn and McCluskey (2025), who proved as well a ``strong threshold theorem", namely that when the DFE is unstable, a second equilibrium which is globally asymptotically stable must exist. Below, we obtain in Theorem \ref{thm:TK_DFE} some results that extend beyond rank one. For example, a nontrivial positive equilibrium exists if and only if the spectral equation $\rho(\widetilde K(S))=1$, admits a strictly positive solution. Also, we showed that Bonzi-Iggidr-Sallet bilinear models with rank one NGM may be classified in two classes, with slight variations in the eigenvector formulas, and that extensions in the presence of feedback from infectious to susceptible are possible. Another takeout from the previous works, which we clarify in a revisit of the seven results in the rank one case, is that Lyapunov functions for both the DFE and the EE may be constructed using as weights the left Perron eigenvector $\pi(S)$ of the \NGM\ (NGM) $K=F(S) V^{-1}$, where $S$ denote all the non-infectious variables, and when a positive ODE leaves a siphon face, it does so along the right Perron eigenvector $w(S)$ of $\T K= V^{-1}F(S)$. The question of whether this continues to be true beyond rank one is explored in ongoing work.

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arXiv2026

A cocktail of chemical reaction networks and mathematical epidemiology tools for positive ODE stability problems

We continue recent attempts to put together concepts and results of Chemical Reaction Networks theory (CRNT) and Mathematical Epidemiology (ME), for solving problems of stability of positive ODEs. We provide first an elegant CRN-flavored generalization of the most cited result in ME, the Next Generation Matrix (NGM) theorem. We review next the "symbolic-numeric approach of Vassena and Stadler, which tackles bifurcation problems by viewing the characteristic polynomial of the Jacobian at fixed points as a formal polynomial in the "symbolic reactivities", and identifies its coefficients as "Child Selection minors of the stoichiometric matrix". We also review two applications of this approach using the Mathematica package Epid-CRN tools from both CRNT and ME.

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arXiv2026

Relay transitions and invasion thresholds in multi-strain rumor models: a chemical reaction network approach

The historical quest for unifying the concepts and methods of Chemical Reaction Networks theory (CRNT), Mahematical Epidemiology (ME) and ecology has received increased attention in the last years and has led in particular to the development of the symbolic package EpidCRN, for automatic analysis of positive ODEs, which implements tools from all these disciplines like siphons, reproduction functions and invasion numbers, Child-Selection expansions, etc. We illustrate below the convenience of using this package on some recent online social network (OSN) rumor spreading models, with emphasis on showing how CRNT throws a new light on their analysis. Specifically, we organise the boundary dynamics via the lattice of invariant faces generated by minimal siphons, and establish that stability transitions take the form of \emph{relays}: for each distance-one cover in the siphon lattice, a single invasion inequality simultaneously governs the loss of transversal stability of the resident equilibrium and the existence of a successor equilibrium on the adjacent face. For the base OSN model ($\omega=0$) all boundary and interior equilibria admit explicit rational formulas, and the relay table is fully verified using invasion numbers computed symbolically by EpidCRN. For the variant with waning spreading impulse ($\omega>0$), the relay structure is analysed via transversal Jacobian blocks; three equilibria involve irrational coordinates and their stability is predicted by the relay framework subject to direct Routh--Hurwitz verification. The relay mechanism is then situated in its normal-form context (siphon-induced transcritical bifurcations), distinguished from classical transcritical bifurcations along four structural axes, and compared with Hofbauer invasion graphs.

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arXiv2025

On the Similarity between Epidemiologic Strains, Minimal Self-Replicable Siphons, and autocatalytic cores in (Chemical) Reaction Networks: Towards a Unifying Framework

We aim to study boundary stability and persistence of positive odes in mathematical epidemiology models by importing structural tools from chemical reaction networks. This is largely a review work, which attempts to bring closer together the fields of mathematical epidemiology (ME), and chemical reaction networks (CRNs), based on several observations. We started by observing the conceptual correspondence between epidemiologic strains and both critical minimal siphons and minimal autocatalytic sets (cores) in an underlying CRN, and confirmed this in all the models we studied. We leverage this to provide a definition of the disease free equilibrium (DFE) face/infected set as the union of either all minimal siphons, or of all cores (they coincide always in our examples). Next, we provide a proposed definition of ME models, as models which have a unique boundary fixed point on the DFE face, and for which the Jacobian of the infected subnetwork admits a regular splitting, which allows defining the famous next generating matrix (NGM). We then define the interaction graph on minimal siphons (IGMS), whose vertices are minimal siphons, and whose edges indicate the existence of reactions producing species in one siphon from species in another. When this graph is acyclic, we say the model exhibits a Acyclic Minimal Siphon Decomposition (AMSD). For AMSD models whose minimal siphons partition the infection species, we show that the NGM is block triangular after permutation, which implies the classical max structure of the reproduction number R0 for multi-strain models. We implement algorithms to compute IGMS and detect AMSD in the Epid-CRN Mathematica package (https://github.com/florinav/EpidCRNmodels) (which contains already modules to identify minimal siphons, criticality, drainability, self-replicability, etc).

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arXiv2025

The Boundary Reproduction Number for Determining Boundary Steady State Stability in Chemical Reaction Systems

We introduce the boundary reproduction number, adapted from the next generation matrix method, to assess whether an infusion of species will persist or become exhausted in a chemical reaction system. Our main contributions are as follows: (a) we show how the concept of a siphon, prevalent in Petri nets and chemical reaction network theory, identifies sets of species that may become depleted at steady state, analogous to a disease-free boundary steady state; (b) we develop an approach for incorporating biochemically motivated conservation laws, which allows the stability of boundary steady states to be determined within specific compatibility classes; and (c) we present an effective heuristic for decomposing the Jacobian of the system that reduces the computational complexity required to compute the stability domain of a boundary steady state. The boundary reproduction number approach significantly simplifies existing parameter-dependent methods for determining the stability of boundary steady states in chemical reaction systems and has implications for the capacity of critical metabolites and substrates in metabolic pathways to become exhausted.

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arXiv2024

Stability in Reaction Network Models via an Extension of the Next Generation Matrix Method

In this essay, we investigate some relations between Chemical Reaction Networks (CRN) and Mathematical Epidemiology (ME) and report on several pleasant surprises which we had simply by putting these two topics together. Firstly, we propose a definition of ME models as a subset of CRN models. Secondly, we review a fundamental stability result for boundary points, known in ME as the NGM method since it replaces the investigation of the Jacobian by that of a matrix whose origins lie in probability (the theory of branching processes). This important result seems to be little known outside of ME; even in ME, it has not been made clear before that the method gets sometimes the right answer, even though the conditions of the NGM theorem are not all satisfied. Thus, beyond the theorem, there is a heuristic approach, the validity conditions for which are not sufficiently understood. Thirdly, we show that some simple CRN models with absolute concentration robustness (ACR), are close qualitatively to simple ME models, in the sense that they have an unique disease free equilibrium, and a unique interior fixed point, and the latter enters the positive domain and becomes stable precisely when $R_0:=s_{dfe} \mathcal{R}=\frac{s_{dfe}}{s_e}>1.$ (where $s$ denotes the "ACR species"). Thus, for these "ME type models", a "relay phenomena" takes place: precisely when the DFE loses stability, a new fixed point enters the domain, and takes over. Last but not least, we offer in the associated GitHub repository https://github.com/adhalanay/epidemiology_crns a Mathematica package, Epid-CRN, which is addressed to researchers of both disciplines, and provide illustrative notebooks, which in particular solve a few minor open ME and CRN problems. Our package may also be used to study easy cases of analogue continuous time Markov chain (CTMC) ME and CRN models.

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arXiv2024

Advancing Mathematical Epidemiology and Chemical Reaction Network Theory via Synergies Between Them

Our paper reviews some key concepts in chemical reaction network theory and mathematical epidemiology, and examines their intersection, with three goals. The first is to make the case that mathematical epidemiology (ME), and also related sciences like population dynamics, virology, ecology, etc., could benefit by adopting the universal language of essentially non-negative kinetic systems as developed by chemical reaction network (CRN) researchers. In this direction, our investigation of the relations between CRN and ME lead us to propose for the first time a definition of ME models, stated in Open Problem 1. Our second goal is to inform researchers outside ME of the convenient next generation matrix (NGM) approach for studying the stability of boundary points, which do not seem suficiently well known. Last but not least, we want to help students and researchers who know nothing about either ME or CRN to learn them quickly, by offering them a Mathematica package "BootCamp", located at https://github.com/adhalanay/epidemiology_crns, including illustrating notebooks (and certain sections below will contain associated suggested notebooks; however, readers with experience may safely skip the bootcamp). We hope that the files indicated in the titles of various sections will be helpful, though of course improvement is always possible, and we ask the help of the readers for that.

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arXiv2024

Advancing Mathematical Epidemic Modeling via synergies with Chemical Reaction Network Theory and Lagrange-Hamilton Geometry

This essay reviews some key concepts in mathematical epidemiology and examines the intersection of this field with related scientific disciplines, such as chemical reaction network theory and Lagrange-Hamilton geometry. Through a synthesis of theoretical insights and practical perspectives, we underscore the significance of essentially non-negative kinetic systems in the development and implementation of robust epidemiological models. Our purpose is to make the case that currently mathematical modeling of epidemiology is focusing too much on simple particular cases, and maybe not enough on more complex models, whose challenges would require cooperation with scientific computing experts and with researchers in the "sister disciplines" involving essentially nonnegative kinetic systems (like virology, ecology, chemical reaction networks, population dynamics, etc).

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arXiv2024

Finding bifurcations in mathematical epidemiology via reaction network methods

Mathematical Epidemiology (ME) shares with Chemical Reaction Network Theory (CRNT) the basic mathematical structure of its dynamical systems. Despite this central similarity, methods from CRNT have been seldom applied to solving problems in ME. We explore here the applicability of CRNT methods to find bifurcations at endemic equilibria of ME models. We adapt three CRNT methods to the features of ME. First, we prove that essentially all ME models admit Hopf bifurcations for certain monotone choices of the interaction functions. Second, we offer a parametrization of equilibria Jacobians of ME systems where few interactions are not in mass action form. Third, for a quite general class of models, we show that periodic oscillations in closed systems imply periodic oscillations when demography is added. Finally, we apply such results to two families of networks: a general SIR model with a nonlinear force of infection and treatment rate and a recent SIRnS model with a gradual increase in infectiousness. We give both necessary conditions and sufficient conditions for the occurrence of bifurcations at endemic equilibria of both families.

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arXiv2023

Algorithmic approach for an unique definition of the next generation matrix

The basic reproduction number R0 is a concept which originated in population dynamics, mathematical epidemiology, and ecology and is closely related to the mean number of children in branching processes.We offer below three new contributions to the literature: 1) We order a universal algorithmic definition of a (F, V) gradient decomposition (and hence of the resulting R0), which requires a minimal input from the user, namely the specification of an admissible set of disease/infection variables. We also present examples where other choices may be more reasonable, with more terms in F, or more terms in V . 2) We glean out from the works of Bacaer a fixed point equation (8) for the extinction probabilities of a stochastic model associated to a deterministic ODE model, which may be expressed in terms of the (F, V ) decomposition. The fact that both R0 and the extinction probabilities are functions of (F, V ) underlines the centrality of this pair, which may be viewed as more fundamental than the famous next generation matrix FV^{-1}. 3) We suggest introducing a new concept of sufficient/minimal disease/infection set (sufficient for determining R0). More precisely, our universal recipe of choosing "new infections" once the "infections" are specified suggests focusing on the choice of the latter, which is also not unique. The maximal choice of choosing all compartments which become 0 at the given boundary point seems to always work, but is the least useful for analytic computations, therefore we propose to investigate the minimal one. As a bonus, this idea seems useful for understanding the Jacobian factorization approach for computing R0 . Last but not least, we offer Mathematica scripts and implement them for a large variety of examples, which illustrate that our recipe others always reasonable results, but that sometimes other reasonable (F, V ) decompositions are available as well.

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arXiv2023

Dynamics of an SIR epidemic model with limited medical resources, revisited and corrected

This paper generalizes and corrects a famous paper (more than 200 citations) concerning Hopf and Bogdanov-Takens bifurcations due to L. Zhou and M. Fan, "Dynamics of an SIR epidemic model with limited medical resources revisited", in which we discovered a significant numerical error. Importantly, unlike the paper of Zhou and Fan and several other papers that followed them, we offer a notebook where the reader may recover all the results and modify them for analyzing similar models. Our calculations lead to the introduction of some interesting symbolic objects, "Groebner eliminated traces and determinants" - see (4.5), (4.6), which seem to have appeared here for the first time and which might be of independent interest. We hope our paper might serve as yet another alarm bell regarding the importance of accompanying papers involving complicated hand computations by electronic notebooks.

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arXiv2023

Explicit mathematical epidemiology results on age renewal kernels and R0 formulas are often consequences of the rank one property of the next generation matrix

A very large class of ODE epidemic models (2.2) discussed in this paper enjoys the property of admitting also an integral renewal formulation, with respect to an "age of infection kernel" a(t) which has a matrix exponential form (3.2). We observe first that a very short proof of this fact is available when there is only one susceptible compartment, and when its associated "new infections" matrix has rank one. In this case, a(t) normalized to have integral 1, is precisely the probabilistic law which governs the time spent in all the "infectious states associated to the susceptible compartment", and the normalization is precisely the basic replacement number. The Laplace transform (LT) of a(t) is a generalization of the basic replacement number, and its structure reflects the laws of the times spent in each infectious state. Subsequently, we show that these facts admit extensions to processes with several susceptible classes, provided that all of them have a new infections matrix of rank one. These results reveal that the ODE epidemic models highlighted below have also interesting probabilistic properties.

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arXiv2022

On a three-dimensional and two four-dimensional oncolytic viro-therapy models

We revisit here and carry out further works on tumor-virotherapy compartmental models of [Tian, 2011, Wang et al., 2013, Phan and Tian, 2017, Guo et al., 2019]. The results of these papers are only slightly pushed further. However, what is new is the fact that we make public our electronic notebooks, since we believe that easy electronic reproducibility is crucial in an era in which the role of the software becomes very important.

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arXiv2022

New results and open questions for SIR-PH epidemic models with linear birth rate, loss of immunity, vaccination, and disease and vaccination fatalities

Our paper presents three new classes of models: SIR-PH, SIR-PH-FA, and SIR-PH-IA, and states two problems we would like to solve about them. Recall that deterministic mathematical epidemiology has one basic general law, the R0 alternative" of [52, 51], which states that the local stability condition of the disease free equilibrium may be expressed as R0 < 1, where R0 is the famous basic reproduction number, which plays also a major role in the theory of branching processes. The literature suggests that it is impossible to find general laws concerning the endemic points. However, it is quite common that 1. When R0 > 1, there exists a unique fixed endemic point, and 2. the endemic point is locally stable when R0 > 1. One would like to establish these properties for a large class of realistic epidemic models (and we do not include here epidemics without casualties). We have introduced in [7, 5] a "simple", but broad class of "SIR-PH models" with varying population, with the express purpose of establishing for these processes the two properties above. Since that seemed still hard, we have introduced a further class of "SIR-PH-FA" models, which may be interpreted as approximations for the SIR-PH models, and which includes simpler models typically studied in the literature (with constant population, without loss of immunity, etc). The goal of our paper is to draw attention to the two open problems above, for the SIR-PH, SIR-PH-FA, and also for a second, more refined "intermediate approximation" SIR-PH-IA. We illustrate the current status-quo by presenting new results on a generalization of the SAIRS epidemic model of [44, 40].

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arXiv2022

On matrix-SIR Arino models with linear birth rate, loss of immunity, disease and vaccination fatalities, and their approximations

In this work we study the stability properties of the equilibrium points of deterministic epidemic models with nonconstant population size. Models with nonconstant population have been studied in the past only in particular cases, two of which we review and combine. Our main result shows that for simple "matrix epidemic models" introduced in [1], an explicit general formula for the reproduction number and the corresponding "weak stability alternative" still holds, under small modifications, for models with nonconstant population size, and even when the model allows for vaccination and loss of immunity. The importance of this result is clear once we note that the models of [1] include a large number of viral and bacterial models of epidemic propagation, including for example the totality of homogeneous COVID-19 models. To better understand the nature of the result, we emphasize that the models proposed in [1] and considered here are extensions of the SIR-PH model, which is essentially characterized by a phase-type distribution that models transitions between the "disease/infectious compartments". In these cases, the reproduction number and a certain Lyapunov function for the disease free equilibrium are explicitly expressible. Not surprisingly, accounting for varying demography, loss of immunity, and vaccinations lead to several challenges. One of the most important is that a varying population size leads to multiple endemic equilibrium points: this is in contrast with "classic models" which in general admit unique disease-free and endemic equilibria. As a special case of our analysis, we consider a "first approximation" (FA) of our model, which coincides with the constant-demography model often studied in the literature, and for which more explicit results are available. Furthermore, we propose a second heuristic approximation named "intermediate approximation" (IA).

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arXiv2022

Stability analysis of an eight parameter SIR-type model including loss of immunity, and disease and vaccination fatalities

We revisit here a landmark five parameter SIR-type model of [DvdD93, Sec. 4], which is maybe the simplest example where a complete picture of all cases, including non-trivial bistability behavior, may be obtained using simple tools. We also generalize it by adding essential vaccination and vaccination-induced death parameters, with the aim of revealing the role of vaccination and its possible failure. The main result is Theorem 5, which describes the stability behavior of our model in all possible cases.

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arXiv2021

Optimal Control of a SIR Epidemic With ICU Constraints and Target Objectives

The aim of this paper is to provide a rigorous mathematical analysis of an optimal control problem with SIR dynamics. The main feature of our study is the presence of state constraints (related to intensive care units ICU capacity) and strict target objectives (related to the immunity threshold). The first class of results provides a comprehensive description of different zones of interest using viability tools. The second achievement is a thorough mathematical analysis of Pontryagin extremals for the aforementioned problem allowing to obtain an explicit closed-loop feedback optimal control. All our theoretical results are numerically illustrated for a further understanding of the geometrical features and scenarios.

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arXiv2021

A review of matrix SIR Arino epidemic models

Many of the models used nowadays in mathematical epidemiology, in particular in COVID-19 research, belong to a certain sub-class of compartmental models whose classes may be divided into three "(x, y, z)" groups, which we will call respectively "susceptible/entrance, diseased, and output" (in the classic SIR case, there is only one class of each type). Roughly, the ODE dynamics of these models contain only linear terms, with the exception of products between x and y terms. It has long been noticed that the basic reproduction number R has a very simple formula (3.3) in terms of the matrices which define the model, and an explicit first integral formula (3.8) is also available. These results can be traced back at least to [ABvdD+07] and [Fen07], respectively, and may be viewed as the "basic laws of SIR-type epidemics"; however many papers continue to reprove them in particular instances (by the next-generation matrix method or by direct computations, which are unnecessary). This motivated us to redraw the attention to these basic laws and provide a self-contained reference of related formulas for (x, y, z) models. We propose to rebaptize the class to which they apply as matrix SIR epidemic models, abbreviated as SYR, to emphasize the similarity to the classic SIR case. For the case of one susceptible class, we propose to use the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution. We note that to each SIR-PH model, one may associate a scalar quantity Y(t) which satisfies "classic SIR relations", see (3.8). In the case of several susceptible classes, this generalizes to (5.10); in a future paper, we will show that (3.8), (5.10) may be used to obtain approximate control policies which compare well with the optimal control of the original model.

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