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Complex network model for COVID-19: human behavior, pseudo-periodic solutions and multiple epidemic waves. (English) Zbl 1501.34046

Summary: We propose a mathematical model for the transmission dynamics of SARS-CoV-2 in a homogeneously mixing non constant population, and generalize it to a model where the parameters are given by piecewise constant functions. This allows us to model the human behavior and the impact of public health policies on the dynamics of the curve of active infected individuals during a COVID-19 epidemic outbreak. After proving the existence and global asymptotic stability of the disease-free and endemic equilibrium points of the model with constant parameters, we consider a family of Cauchy problems, with piecewise constant parameters, and prove the existence of pseudo-oscillations between a neighborhood of the disease-free equilibrium and a neighborhood of the endemic equilibrium, in a biologically feasible region. In the context of the COVID-19 pandemic, this pseudo-periodic solutions are related to the emergence of epidemic waves. Then, to capture the impact of mobility in the dynamics of COVID-19 epidemics, we propose a complex network with six distinct regions based on COVID-19 real data from Portugal. We perform numerical simulations for the complex network model, where the objective is to determine a topology that minimizes the level of active infected individuals and the existence of topologies that are likely to worsen the level of infection. We claim that this methodology is a tool with enormous potential in the current pandemic context, and can be applied in the management of outbreaks (in regional terms) but also to manage the opening/closing of borders.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92C60 Medical epidemiology
92D30 Epidemiology
92B20 Neural networks for/in biological studies, artificial life and related topics
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
37C60 Nonautonomous smooth dynamical systems
34C25 Periodic solutions to ordinary differential equations
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[1] Ambrosio, Benjamin; Aziz-Alaoui, M. A., On a coupled time-dependent SIR models fitting with New York and New-Jersey states COVID-19 data, Biology, 9, 6, 135 (2020)
[2] Aziz-Alaoui, M. A., Synchronization of chaos, (Françoise, Jean-Pierre; Naber, Gregory L.; Tsun, Tsou Sheung, Encyclopedia of Mathematical Physics (2006), Academic Press: Academic Press Oxford), 213-226 · Zbl 1170.00001
[3] Banos, Arnaud; Corson, Nathalie; Gaudou, Benoit; Laperrière, Vincent; Coyrehourcq, Sébastien Rey, The importance of being hybrid for spatial epidemic models: a multi-scale approach, Systems, 3, 4, 309-329 (2015)
[4] Belykh, Vladimir N.; Belykh, Igor V.; Hasler, Martin, Connection graph stability method for synchronized coupled chaotic systems, Phys. D: Nonlinear Phenom., 195, 1-2, 159-187 (2004) · Zbl 1098.82622
[5] Benlloch, José-María; Cortés, Juan-Carlos; Martínez-Rodríguez, David; Julián, Raul-S.; Villanueva, Rafael-J., Effect of the early use of antivirals on the COVID-19 pandemic. A computational network modeling approach, Chaos Solitons Fractals, 140, Article 110168 pp. (2020) · Zbl 07508237
[6] Cantin, Guillaume, Non identical coupled networks with a geographical model for human behaviors during catastrophic events, Int. J. Bifurc. Chaos, 27, 14, Article 1750213 pp. (2017) · Zbl 1382.34055
[7] Cantin, Guillaume; Silva, Cristiana J., Influence of the topology on the dynamics of a complex network of HIV/AIDS epidemic models, AIMS Math., 4, 1145 (2019) · Zbl 1484.92101
[8] Centers for Disease Control and Prevention, Coronavirus disease 2019 (COVID-19), 2020. · Zbl 07505139
[9] Denu, Dawit; Ngoma, Sedar; Salako, Rachidi B., Existence of traveling wave solutions of a deterministic vector-host epidemic model with direct transmission, J. Math. Anal. Appl., 487, 1, Article 123995 pp. (2020) · Zbl 1435.92069
[10] Direção Geral da Saúde COVID-19, Ponto de situação atual em Portugal, 2020.
[11] Epstein, Joshua M.; Parker, Jon; Cummings, Derek; Hammond, Ross A., Coupled contagion dynamics of fear and disease: mathematical and computational explorations, PLoS ONE, 3, 12, Article e3955 pp. (2008)
[12] Euronews, Coronavirus second wave: which countries in Europe are experiencing a fresh spike in COVID-19 cases?, 2020.
[13] European Centre for Disease Prevention and Control, Guidelines for the implementation of non-pharmaceutical interventions against COVID-19, 2020.
[14] Golubitsky, Martin; Stewart, Ian, Nonlinear dynamics of networks: the groupoid formalism, Bull. Am. Math. Soc., 43, 3, 305-364 (2006) · Zbl 1119.37036
[15] Hsu, Cheng-Hsiung; Lin, Jian-Jhong, Stability of traveling wave solutions for nonlinear cellular neural networks with distributed delays, J. Math. Anal. Appl., 470, 1, 388-400 (2019) · Zbl 1432.34091
[16] Krantz, Steven G.; Rao, Arni S. R. Srinivasa, Level of underreporting including underdiagnosis before the first peak of COVID-19 in various countries: preliminary retrospective results based on wavelets and deterministic modeling, Infect. Control Hosp. Epidemiol., 1-3 (2020)
[17] LaSalle, Joseph, Some extensions of Liapunov’s second method, IRE Trans. Circuit Theory, 7, 4, 520-527 (1960)
[18] Moradian, N.; Ochs, H. D.; Sedikies, C., The urgent need for integrated science to fight COVID-19 pandemic and beyond, J. Transl. Med., 205 (2020)
[19] Ndaïrou, Faïçal; Area, Iván; Nieto, Juan J.; Torres, Delfim F. M., Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan, Chaos Solitons Fractals, 135, Article 109846 pp. (2020) · Zbl 1489.92171
[20] Perko, Lawrence, Differential Equations and Dynamical Systems, vol. 7 (2013), Springer Science & Business Media · Zbl 0973.34001
[21] Pordata, Taxa de crescimento anual médio segundo os Censos (
[22] Sarkar, Kankan; Khajanchi, Subhas; Nieto, Juan J., Modeling and forecasting the COVID-19 pandemic in India, Chaos Solitons Fractals, 139, Article 110049 pp. (2020) · Zbl 07505072
[23] Silva, Cristiana J.; Cruz, Carla; Torres, Delfim F. M.; Muñuzuri, Alberto P.; Carballosa, Alejandro; Area, Ivan; Nieto, Juan J.; Fonseca-Pinto, Rui; Passadouro da Fonseca, Rui; Soares dos Santos, Estevão; Abreu, Wilson; Mira, Jorge, Optimal control of the COVID-19 pandemic: controlled sanitary deconfinement in Portugal, Sci. Rep., 11 (2021), Art. 3451, 15 pp.
[24] Smith, Hal L.; Thieme, Horst R., Dynamical Systems and Population Persistence, vol. 118 (2011), American Mathematical Soc. · Zbl 1214.37002
[25] Smoller, Joel, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, vol. 258 (1994), Springer-Verlag: Springer-Verlag New York · Zbl 0807.35002
[26] van den Driessche, Pauline; Watmough, James, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 1-2, 29-48 (2002) · Zbl 1015.92036
[27] Vinceti, Marco; Filippini, Tommaso; Rothman, Kenneth J.; Ferrari, Fabrizio; Goffi, Alessia; Maffeis, Giuseppe; Orsini, Nicola, Lockdown timing and efficacy in controlling COVID-19 using mobile phone tracking, EClinicalMedicine, 25, Article 100457 pp. (2020)
[28] Wang, Zhi-Cheng; Wu, Jianhong, Traveling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. A, Math. Phys. Eng. Sci., 466, 2113, 237-261 (2010) · Zbl 1195.35291
[29] Word Health Organization, WHO announces COVID-19 outbreak a pandemic, 2020.
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