A Markovian model for the spread of the SARS-CoV-2 virus. (English) Zbl 1521.92093
Authors’ abstract: We propose a Markovian stochastic approach to model the spread of a SARS-CoV-2-like infection within a closed group of humans. The model takes the form of a partially observable Markov decision process (POMDP), whose states are given by the number of subjects in different health conditions. The model also exposes the different parameters that have an impact on the spread of the disease and the various decision variables that can be used to control it (e.g, social distancing, number of tests administered to single out infected subjects). The model describes the stochastic phenomena that underlie the spread of the epidemic and captures, in the form of deterministic parameters, some fundamental limitations in the availability of resources (hospital beds and test swabs). The model lends itself to different uses. For a given control policy, it is possible to verify if it satisfies an analytical property on the stochastic evolution of the state (e.g., to compute probability that the hospital beds will reach a fill level, or that a specified percentage of the population will die). If the control policy is not given, it is possible to apply POMDP techniques to identify an optimal control policy that fulfills some specified probabilistic goals. Whilst the paper primarily aims at the model description, we show with numeric examples some of its potential applications.
Reviewer: Jiaying Zhou (Shenzhen)
Keywords:
modelling and control of biomedical systems; healthcare management; stochastic systems; SARS-CoV-2; virusReferences:
| [1] | Ahmadi, M.; Jansen, N.; Wu, B.; Topcu, U., Control theory meets pomdps: A hybrid systems approach, IEEE Transactions on Automatic Control (2020) |
| [2] | Allen, L.; Brauer, F.; van den Driessche, P.; Bauch, C.; Wu, J.; Castillo-Chavez, C., Mathematical epidemiology, (Lecture notes in mathematics (2008), Springer Berlin Heidelberg) · Zbl 1159.92034 |
| [3] | Anderson, R.; May, R., Infectious diseases of humans: Dynamics and control, (Dynamics and control (1992), OUP Oxford) |
| [4] | Baier, C.; Katoen, J., Principles of model checking (2008), MIT Press · Zbl 1179.68076 |
| [5] | Bernoulli, D., Essai d’une nouvelle analyse de la mortalite causee par la petite ve- role, et des avantage de l’inoculation pour la prevenir, Memoirs Physics Academy Royal Science, 1, 6 (1760) |
| [6] | Blanchini, F.; Franco, E.; Giordano, G.; Mardanlou, V.; Montessoro, P. L., Compartmental flow control: Decentralization, robustness and optimality, Automatica, 64, 18-28 (2016) · Zbl 1329.93007 |
| [7] | Brauer, F.; Castillo-Chavez, C.; Feng, Z., Mathematical models in epidemiology. Texts in applied mathematics (2019), Springer: Springer New York · Zbl 1433.92001 |
| [8] | Browne, M. C.; Clarke, E. M.; Grümberg, O., Characterizing finite kripke structures in propositional temporal logic, Theoretical Computer Science, 59, 1-2, 115-131 (1988) · Zbl 0677.03011 |
| [9] | Cassandras, C. G.; Lafortune, S., Introduction to discrete event systems (2009), Springer Science & Business Media |
| [10] | Chauhan, K., Epidemic analysis using traditional model checking and stochastic simulation (2015), IIT Hyderabad, [Master’s thesis] |
| [11] | Dobay, M. P.; Dobay, A.; Bantang, J.; Mendoza, E., How many trimers? modeling influenza virus fusion yields a minimum aggregate size of six trimers, three of which are fusogenic, Mol. BioSyst., 7, 2741-2749 (2011) |
| [12] | Feinberg, E. A.; Shwartz, A., Handbook of Markov decision processes: Methods and applications (2012), Springer Science & Business Media · Zbl 0979.90001 |
| [13] | Forejt, V.; Kwiatkowska, M.; Norman, G.; Parker, D., Automated verification techniques for probabilistic systems, (Bernardo, M.; Issarny, V., Formal methods for eternal networked software systems. Formal methods for eternal networked software systems, LNCS, vol. 6659 (2011), Springer), 53-113 |
| [14] | Gani, J.; Jerwood, D., Markov chain methods in chain binomial epidemic models, Biometrics, 591-603 (1971) |
| [15] | Ghezzi, L. L.; Piccardi, C., Pid control of a chaotic system: An application to an epidemiological model, Automatica, 33, 2, 181-191 (1997) · Zbl 0872.93060 |
| [16] | Giordano, G.; Blanchini, F.; Bruno, R.; Colaneri, P.; Di Filippo, A.; Di Matteo, A., Modelling the covid-19 epidemic and implementation of population-wide interventions in italy, Nature Medicine, 1-6 (2020) |
| [17] | Haksar, R. N.; Schwager, M., Controlling large, graph-based mdps with global control capacity constraints: An approximate lp solut, (2018 IEEE conference on decision and control (2018)), 35-42 |
| [18] | Hansson, H.; Jonsson, B., A logic for reasoning about time and reliability, Formal Aspects of Computing, 6, 5, 512-535 (1994) · Zbl 0820.68113 |
| [19] | Ji, C.; Jiang, D.; Yang, Q.; Shi, N., Dynamics of a multigroup sir epidemic model with stochastic perturbation, Automatica, 48, 1, 121-131 (2012) · Zbl 1244.93154 |
| [20] | Kaelbling, L. P.; Littman, M. L.; Cassandra, A. R., Planning and acting in partially observable stochastic domains, Artificial Intelligence, 101, 1-2, 99-134 (1998) · Zbl 0908.68165 |
| [21] | Kermack, W. O.; McKendrick, A. G.; Walker, G. T., A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London. Series A, Containing Papers of A Mathematical and Physical Character, 115, 772, 700-721 (1927) · JFM 53.0517.01 |
| [22] | Khanafer, A.; Başar, T.; Gharesifard, B., Stability of epidemic models over directed graphs: A positive systems approach, Automatica, 74, 126-134 (2016) · Zbl 1348.93226 |
| [23] | Kwiatkowska, M.; Norman, G.; Parker. Prism, D., 4.0: Verification of probabilistic real-time systems, (International conference on computer aided verification (2011), Springer), 585-591 |
| [24] | Liu, H.; Yang, Q.; Jiang, D., The asymptotic behavior of stochastically perturbed di sir epidemic models with saturated incidences, Automatica, 48, 5, 820-825 (2012) · Zbl 1246.93117 |
| [25] | Nasir, A.; Baig, H. R.; Rafiq, M., Epidemics control model with consideration of seven-segment population model, SN Applied Sciences, 2, 10, 1-9 (2020) |
| [26] | Razzaq, M.; Ahmad, J., Petri net and probabilistic model checking based approach for the modelling, simulation and verification of internet worm propagation, PLOS ONE, 10, 12, 1-22 (2016) |
| [27] | Riccardo, F.; Ajelli, M.; Andrianou, X. D., Epidemiological characteristics of Covid-19 cases and estimates of the reproductive numbers 1 month into the epidemic, Italy, 28 January To 31 March 2020 (2020), Eurosurveillance |
| [28] | Roveri, M.; Ivankovic, F.; Palopoli, L.; Fontanelli, D., Verifying a stochastic model for the spread of a SARS-CoV-2-like infection: opportunities and limitations, (Int. conf. of the Italian association for artificial intelligence (AIxIA). Int. conf. of the Italian association for artificial intelligence (AIxIA), LNAI (2023), Springer), To appear |
| [29] | Sattenspiel, L., Modeling the spread of infectious disease in human populations, American Journal of Physical Anthropology, 33, S11, 245-276 (1990) |
| [30] | Tuckwell, H. C.; Williams, R. J., Some properties of a simple stochastic epidemic model of sir type, Mathematical Biosciences, 208, 1, 76-97 (2007) · Zbl 1116.92061 |
| [31] | Viet, A.-F.; Krebs, S.; Rat-Aspert, O.; Jeanpierre, L.; Belloc, C.; Ezanno, P., A modelling framework based on mdp to coordinate farmers’ disease control decisions at a regional scale, PLOS ONE, 13, 6, 1-20 (2018) |
| [32] | Yousefpour, A.; Jahanshahi, H.; Bekiros, S., Optimal policies for control of the novel coronavirus disease (covid-19) outbreak, Chaos, Solitons & Fractals, 136, Article 109883 pp. (2020) |
| [33] | Zardini, A.; Galli, M.; Tirani, M.; Cereda, D.; Manica, M.; Trentini, F., A quantitative assessment of epidemiological parameters required to investigate covid-19 burden, Epidemics, 37, Article 100530 pp. (2021) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
