Basnarkov, Lasko

Preferred name

Basnarkov, Lasko

Official Name

Basnarkov, Lasko

Main Affiliation

lasko.basnarkov@finki.ukim.mk

Now showing 1 - 10 of 43

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Algorithmic Approach for a Unique Definition of the Next-Generation Matrix
(MDPI, 2023-12-21)
Avram, Florin
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Adenane, Rim
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Johnston, Matthew D
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 (reflecting the fact that the phenomena of interest are well approximated via branching processes, at their inception). Despite the very extensive literature around 𝑅0 for deterministic epidemic models, we believe there are still aspects which are not fully understood. Foremost is the fact that 𝑅0 is not a function of the original ODE model, unless we also include in it a certain (𝐹,𝑉) gradient decomposition, which is not unique. This is related to the specification of the “infected compartments”, which is also not unique. A second interesting question is whether the extinction probabilities of the natural continuous time Markovian chain approximation of an ODE model around boundary points (disease-free equilibrium and invasion points) are also related to the (𝐹,𝑉) gradient decomposition. We offer below several new contributions to the literature: (1) A universal algorithmic definition of a (𝐹,𝑉) gradient decomposition (and hence of the resulting 𝑅0). (2) A fixed point equation 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. Last but not least, we offer Mathematica scripts and implement them for a large variety of examples, which illustrate that our recipe offers always reasonable results, but that sometimes other reasonable (𝐹,𝑉) decompositions are available as well.
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Some Probabilistic Interpretations Related to the Next-Generation Matrix Theory: A Review with Examples.
(MDPI, 2024-08-01)
Avram, Florin
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Adenane, Rim
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The fact that the famous basic reproduction number 𝑅0, i.e., the largest eigenvalue of the next generation matrix 𝐹𝑉−1, sometimes has a probabilistic interpretation is not as well known as it deserves to be. It is well understood that half of this formula, −𝑉, is a Markovian generating matrix of a continuous-time Markov chain (CTMC) modeling the evolution of one individual on the compartments. It has also been noted that the not well-enough-known rank-one formula for 𝑅0 of Arino et al. (2007) may be interpreted as an expected final reward of a CTMC, whose initial distribution is specified by the rank-one factorization of F. Here, we show that for a large class of ODE epidemic models introduced in Avram et al. (2023), besides the rank-one formula, we may also provide an integral renewal representation of 𝑅0 with respect to explicit “age kernels” 𝑎(𝑡), which have a matrix exponential form.This latter formula may be also interpreted as an expected reward of a probabilistic continuous Markov chain (CTMC) model. Besides the rather extensively studied rank one case, we also provide an extension to a case with several susceptible classes.
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An Age of Infection Kernel, an ℛ Formula, and Further Results for Arino–Brauer A, B Matrix Epidemic Models with Varying Populations, Waning Immunity, and Disease and Vaccination Fatalities
(MDPI, 2023-03-08)
Avram, Florin
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Adenane, Rim
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Bianchin, Gianluca
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Goreac, Dan
In this work, we first introduce a class of deterministic epidemic models with varying populations inspired by Arino et al. (2007), the parameterization of two matrices, demography, the waning of immunity, and vaccination parameters. Similar models have been focused on by Julien Arino, Fred Brauer, Odo Diekmann, and their coauthors, but mostly in the case of “closed populations” (models with varying populations have been studied in the past only in particular cases, due to the difficulty of this endeavor). Our Arino–Brauer models contain SIR–PH models of Riano (2020), which are characterized by the phase-type distribution (𝛼⃗ ,𝐴), modeling transitions in “disease/infectious compartments”. The A matrix is simply the Metzler/sub-generator matrix intervening in the linear system obtained by making all new infectious terms 0. The simplest way to define the probability row vector 𝛼⃗ is to restrict it to the case where there is only one susceptible class 𝗌, and when matrix B (given by the part of the new infection matrix, with respect to 𝗌) is of rank one, with 𝐵=𝑏𝛼⃗ . For this case, the first result we obtained was an explicit formula (12) for the replacement number (not surprisingly, accounting for varying demography, waning immunity and vaccinations led to several nontrivial modifications of the Arino et al. (2007) formula). The analysis of (𝐴,𝐵) Arino–Brauer models is very challenging. As obtaining further general results seems very hard, we propose studying them at three levels: (A) the exact model, where only a few results are available—see Proposition 2; and (B) a “first approximation” (FA) of our model, which is related to the usually closed population model often studied in the literature. Notably, for this approximation, an associated renewal function is obtained in (7); this is related to the previous works of Breda, Diekmann, Graaf, Pugliese, Vermiglio, Champredon, Dushoff, and Earn. (C) Finally, we propose studying a second heuristic “intermediate approximation” (IA). Perhaps our main contribution is to draw attention to the importance of (𝐴,𝐵) Arino–Brauer models and that the FA approximation is not the only way to tackle them. As for the practical importance of our results, this is evident, once we observe that the (𝐴,𝐵) Arino–Brauer models include a large number of epidemic models (COVID, ILI, influenza, illnesses, etc.).
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Random walks on networks with centrality-based stochastic resetting
(MDPI, 2023-02-04)
Zelenkovski, Kiril
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Sandev, Trifce
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Metzler, Ralf
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Kocarev, Ljupco
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We introduce a refined way to diffusely explore complex networks with stochastic resetting where the resetting site is derived from node centrality measures. This approach differs from previous ones, since it not only allows the random walker with a certain probability to jump from the current node to a deliberately chosen resetting node, rather it enables the walker to jump to the node that can reach all other nodes faster. Following this strategy, we consider the resetting site to be the geometric center, the node that minimizes the average travel time to all the other nodes. Using the established Markov chain theory, we calculate the Global Mean First Passage Time (GMFPT) to determine the search performance of the random walk with resetting for different resetting node candidates individually. Furthermore, we compare which nodes are better resetting node sites by comparing the GMFPT for each node. We study this approach for different topologies of generic and real-life networks. We show that, for directed networks extracted for real-life relationships, this centrality focused resetting can improve the search to a greater extent than for the generated undirected networks. This resetting to the center advocated here can minimize the average travel time to all other nodes in real networks as well. We also present a relationship between the longest shortest path (the diameter), the average node degree and the GMFPT when the starting node is the center. We show that, for undirected scale-free networks, stochastic resetting is effective only for networks that are extremely sparse with tree-like structures as they have larger diameters and smaller average node degrees. For directed networks, the resetting is beneficial even for networks that have loops. The numerical results are confirmed by analytic solutions. Our study demonstrates that the proposed random walk approach with resetting based on centrality measures reduces the memoryless search time for targets in the examined network topologies.
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Estimation of the basic reproduction number of COVID-19 from the incubation period distribution
(2022)
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Tomovski, Igor
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Avram, Florin
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Cooperation dynamics of generalized reciprocity in state-based social dilemmas
(American Physical Society, 2018-05-14)
Stojkoski, Viktor
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Utkovski, Zoran
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Kocarev, Ljupcho
We introduce a framework for studying social dilemmas in networked societies where individuals follow a simple state-based behavioral mechanism based on generalized reciprocity, which is rooted in the principle “help anyone if helped by someone”. Within this general framework, which applies to a wide range of social dilemmas including, among others, public goods, donation and snowdrift games, we study the cooperation dynamics on a variety of complex network examples. By interpreting the studied model through the lenses of nonlinear dynamical systems, we show that cooperation through generalized reciprocity always emerges as the unique attractor in which the overall level of cooperation is maximized, while simultaneously exploitation of the participating individuals is prevented. The analysis elucidates the role of the network structure, here captured by a local centrality measure which uniquely quantifies the propensity of the network structure to cooperation by dictating the degree of cooperation displayed both at microscopic and macroscopic level. We demonstrate the applicability of the analysis on a practical example by considering an interaction structure that couples a donation process with a public goods game.
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On matrix-SIR Arino models with linear birth rate, loss of immunity, disease and vaccination fatalities, and their approximations
(2021-12-07)
Avram, Florin
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Adenane, Rim
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Bianchin, Gianluca
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Goreac, Dan
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 R and the corresponding “weak stability alternative” [2, Thm 1] 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 [3], which is essentially characterized by a phase-type distribution (~α, A) that models transitions between the “disease/infectious compartments”. In these cases, the reproduction number R and a certain Lyapunov function for the disease free equilibrium are explicitly expressible in terms of (~α, A). 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). We hope that more light on varying population models with loss of immunity and vaccination, which have been largely avoided until now – see though [4–11] – will emerge in the future.
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Random walk with memory on complex networks
(American Physical Society, 2020-10-30)
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Kocarev, Ljupcho
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We study random walk on complex networks with transition probabilities which depend on the current and previously visited nodes. By using an absorbing Markov chain we derive an exact expression for the mean first passage time between pairs of nodes, for a random walk with a memory of one step. We have analyzed one particular model of random walk, where the transition probabilities depend on the number of paths to the second neighbors. The numerical experiments on paradigmatic complex networks verify the validity of the theoretical expressions, and also indicate that the flattening of the stationary occupation probability accompanies a nearly optimal random search.
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Comparative Study of Random Walks with One-Step Memory on Complex Networks
(Springer, Cham, 2023-03-30)
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We investigate searching efficiency of different kinds of random walk on complex networks which rely on local information and one-step memory. For the studied navigation strategies we obtained theoretical and numerical values for the graph mean first passage times as an indicator for the searching efficiency. The experiments with generated and real networks show that biasing based on inverse degree, persistence and local two-hop paths can lead to smaller searching times. Moreover, these biasing approaches can be combined to achieve a more robust random search strategy. Our findings can be applied in the modeling and solution of various real-world problems.
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PageRank approach to ranking national football teams
(2015-04-03)
Lazova, Verica
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The Football World Cup as world’s favorite sporting event is a source of both entertainment and overwhelming amount of data about the games played. In this paper we analyse the available data on football world championships since 1930 until today. Our goal is to rank the national teams based on all matches during the championships. For this purpose, we apply the PageRank with restarts algorithm to a graph built from the games played during the tournaments. Several statistics such as matches won and goals scored are combined in different metrics that assign weights to the links in the graph. Finally, our results indicate that the Random walk approach with the use of right metrics can indeed produce relevant rankings comparable to the FIFA official all-time ranking board.

Basnarkov, Lasko

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