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Florin Avram

Florin Avram appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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math.PRmath.OCPopulations and Evolutionmath.CAmath.DSq-bio.MN

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Published work

12 published item(s)

preprint2026arXiv

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.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

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].

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

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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preprint2022arXiv

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.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Optimizing dividends and capital injections limited by bankruptcy, and practical approximations for the Cramér-Lundberg process

The recent papers Gajek-Kucinsky(2017) and Avram-Goreac-Li-Wu(2020) investigated the control problem of optimizing dividends when limiting capital injections stopped upon bankruptcy. The first paper works under the spectrally negative Lévy model; the second works under the Cramér-Lundberg model with exponential jumps, where the results are considerably more explicit. The current paper has three purposes. First, it illustrates the fact that quite reasonable approximations of the general problem may be obtained using the particular exponential case studied in Avram-Goreac-Li-Wu(2020). Secondly, it extends the results to the case when a final penalty $P$ is taken into consideration as well besides a proportional cost $k>1$ for capital injections. This requires amending the "scale and Gerber-Shiu functions" already introduced in Gajek-Kucinsky(2017). Thirdly, in the exponential case, the results will be made even more explicit by employing the Lambert-W function. This tool has particular importance in computational aspects and can be employed in theoretical aspects such as asymptotics.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

On fluctuation theory for spectrally negative Levy processes with Parisian reflection below, and applications

As well known, all functionals of a Markov process may be expressed in terms of the generator operator, modulo some analytic work. In the case of spectrally negative Markov processes however, it is conjectured that everything can be expressed in a more direct way using the $W$ scale function which intervenes in the two-sided first passage problem, modulo performing various integrals. This conjecture arises from work on Levy processes \cite{AKP,Pispot,APP,Iva,IP, ivanovs2013potential,AIZ,APY}, where the $W$ scale function has explicit Laplace transform, and is therefore easily computable; furthermore it was found in the papers above that a second scale function $Z$ introduced in \cite{AKP} greatly simplifies first passage laws, especially for reflected processes. This paper gathers a collection of first passage formulas for spectrally negative Parisian Lévy processes, expressed in terms of $W,Z$ which may serve as an "instruction kit" for computing quantities of interest in applications, for example in risk theory and mathematical finance. To illustrate the usefulness of our list, we construct a new index for the valuation of financial companies modeled by spectrally negative Lévy processes, based on a Dickson-Waters modifications of the de Finetti optimal expected discounted dividends objective. We offer as well an index for the valuation of conglomerates of financial companies. An implicit question arising is to investigate analog results for other classes of spectrally negative Markovian processes.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

On the central management of risk networks

We introduce a family of risk networks composed from a) several subsidiary branches $U_i(t), i=1,...,I$ necessary for coping with different types of risks, which must all be kept above $0$, and b) a central branch (CB) which bails out the subsidiaries whenever necessary. Ruin occurs when the central branch is ruined. We find out that with one subsidiary ($I=1$), the finite time ruin probability of the central branch may be explicitly written out in terms of the finite time ruin probability of the subsidiary, provided that the CB in the absence of subsidiary bailouts is a deterministic drift. To study other problems, like for example the optimization of dividends to the CB with one subsidiary over a barrier, it is convenient to restrict to the case of phase-type claims to the subsidiary, and study the Markovian phase process at the moments when the CB process reaches new minima. The resulting structure is quite close to that of the phase of a PH/G/1 queue at the moments when it reaches new minima, and this yields in principle numeric approaches to several problems, based on the iterative calculation of the Laplace transforms of the upwards and downwards ladder times (also called excursions, and busy periods). In this work we propose a different approach of further providing matrix exponential approximations for the distributions of the ladder times. The advantage of this approach is that once a SNMAP approximation is obtained, many similar problems may be solved just by applying the recently developed scale matrix methodology. A numeric experiment for the CB barrier optimization problem is provided.

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preprint2014arXiv

Semi-infinite optimization with sums of exponentials via polynomial approximation

We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by low-degree polynomials. Thus, the constraints can be approximated with polynomial inequalities that can be implemented with linear matrix inequalities. Convexity is preserved, but the problem has now a finite number of constraints. We show how to take advantage of the properties of the exponential function in order to build quickly accurate approximations. The problem used for illustration is the least-squares fitting of a positive sum of exponentials to an empirical density. When the exponents are given, the problem is convex, but we also give a procedure for optimizing the exponents. Several examples show that the method is flexible, accurate and gives better results than other methods for the investigated problems.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

On matrix exponential approximations of the infimum of a spectrally negative Levy process

We recall four open problems concerning constructing high-order matrix-exponential approximations for the infimum of a spectrally negative Levy process (with applications to first-passage/ruin probabilities, the waiting time distribution in the M/G/1 queue, pricing of barrier options, etc). On the way, we provide a new approximation, for the perturbed Cramer-Lundberg model, and recall a remarkable family of (not minimal order) approximations of Johnson and Taaffe, which fit an arbitrarily high number of moments, greatly generalizing the currently used approximations of Renyi, De Vylder and Whitt-Ramsay. Obtaining such approximations which fit the Laplace transform at infinity as well would be quite useful.

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