Additive multiple contacts and saturation phenomena in epidemiological models are not detected by R0

José Geiser Villavicencio-Pulido¹ ; Ignacio Barradas² ; Claudia Nila-Luévano³

¹ Universidad Autónoma Metropolitana, Av. Hidalgo Poniente No. 46, Col. La Estación, 52006, Lerma de Villada, Estado de México, Mexico
² Centro de Investigación en Matemáticas, 36023 Guanajuato, Guanajuato, Mexico
³ Independent researcher

Mathematical modelling of natural phenomena, Tome 19 (2024), article no. 8

Résumé

Many infections are transmitted by direct contacts. Usually, one single direct contact is needed to transmit the required minimum infectious load. Most models describe contagions by single contacts using a term of the type mass action law. However, modelling infections that are transmitted after the susceptible individual had contact with several sources of infection requires more than mass action law terms. We call additive multiple contacts those that do not produce infection by themselves, but can produce infection if they happen simultaneously. We are interested in understanding the role played by R0 missing the mark in infections in which the minimum infectious load is reached not only by single contacts but also by additive multiple contacts. We propose different mathematical models describing not only infections by one single contact but also by additive multiple contacts. We show that all models have the same value of R0, but correspond to different epidemiological mechanisms. Two models show contagions by additive multiple contacts and a third one shows reduction of infections by some saturation process which is not captured by R0. This shows that trying to control the epidemics by controlling R0 could be unsufficient or, in some cases, waste resources.

Détail

DOI : 10.1051/mmnp/2024006

Bibliographie

[1] K.S. Crump, D.G. Hoel, C.H. Langley, R. Peto Fundamental carcinogenic processes and their implications for low dose risk assessment Cancer Res. 1976 2973 2979

[2] D.L. Sewell Laboratory-associated infections and biosafety Clin. Microbiol. Rev. 1995 389 405

[3] S. Karimzadeh, R. Bhopal, H. Nguyen Tien Review of infective dose, routes of transmission and outcome of COVID-19 caused by the SARS-COV-2: comparison with other respiratory viruses Epidemiol. Infect. 2021 1 8

[4] S. Basu, Close-range exposure to a COVID-19 carrier: transmission trends in the respiratory tract and estimation of infectious dose. medRxiv (2020).

[5] S. Yezli, J.A. Otter Minimum infective dose of the major human respiratory and enteric viruses transmitted through food and the environment Food Environ. Virol. 2011 1 30

[6] G. Bagheri, B. Rhiede, B. Hejazi, O. Schlenczek, E. Bodenschatz An upper bound on one-to-one exposure to infectious human respiratory particles PNAS 2021 e2110117118

[7] Wm. Liu, S.A. Levin, Y. Iwasa Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models J. Math. Biol. 1986 187 204

[8] P. Van Den Driessche, J. Watmough A simple SIS epidemic model with a backward bifurcation J. Math. Biol. 2000 525 540

[9] B. Buonomo, D. Lacitignola Forces of infection allowing for backward bifurcation in an epidemic model with vaccination and treatment Acta Appl Math 2012 283 293

[10] B. Buonomo, D. Lacitignola On the dynamics of an SEIR epidemic model with a convex incidence rate Ricerche mat. 2008 261 281

[11] V. Capasso, G. Serio A generalization of the Kermack–McKendrick deterministic epidemic model Math. Biosci. 1978 41 61

[12] C. Paulo, M. Correira-Neves, T. Domingos, A.G. Murta, J. Pedrosa Influenza infectious dose may explain the high mortality of the second and third wave of 1918–1919 influenza pandemic PLos One 2010 e11655

[13] C.M. Kribs-Zaleta, J.X. Velasco-Hernandez A simple vaccination model with multiple endemic states Math. Biosci. 2000 183 201

[14] A. Tang, Z.D. Tong, H.L. Wang, Y.X. Dai, K.F. Li, J.N. Liu, W.J. Wu, C. Yuan, M.L. Yu, P. Li, J.B. Yan Detection of novel coronavirus by RT_PCR in stool specimen from asymptomatic child, China Emerg. Infect. Dis. 2020 1337 1339

[15] Z. Hu, C. Song, C. Xu, G. Jin, Y. Chen, X. Xu, H. Ma, W. Chen, Y. Lin, Y. Zheng, J. Wang, Z. Hu, Y. Yi, H. Shen Clinical characteristics of 24 asymptomatic infections with COVID-19 screened among close contacts in Nanjing, China Sci. China Life Sci. 2020 706 711

[16] J. Zhou, Y. Tan, D. Li, X. He, T. Yuan, Y. Long Observation and analysis of 26 cases of asymptomatic SARS-COV2 infection J. Infect. 2020 e69 e70

[17] J. Cai, W. Sun, J. Huang, M. Gamber, J. Wu, G. He Indirect virus transmission in cluster of COVID-19 cases, Wenzhou, China, 2020 Emerg. Infect. Dis. 2020 1343 1345

[18] C. Yang, J. Wang A mathematical model for the novel coronavirus epidemic in Wuhan, China Math. Biosci. Eng. 2020 2708 2724

[19] Y. Jin, W. Wang, S. Xiao A SIRS model with a nonlinear incidence Chaos Solitons Fractals 2007 1482 1497

[20] P. van den Driessche and J. Watmough, Epidemic solutions and endemic catastrophies, in Dynamical Systems and Their Applications in Biology, Vol. 36. American Mathematical Society, Providence (2003) 247–257.

[21] O. Diekmann, J.A.P. Heesterbeek, J.A. Metz On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations J. Math. Biol. 1990 365 382

[22] P. Van Den Driessche, J. Watmough Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission Math. Biosci. 2002 29 48

[23] Ministry of Health, Mexico, http://datosabiertos.salud.gob.mx/gobmx/salud/datos_abiertos/datos_abiertos_covid19.zip.

[24] J.A. Christen, C. Fox A general purpose sampling algorithm for continuous distribution (the t-walk) Bayesian Anal. 2010 263 281

[25] A. Svensson A note on generation times in epidemic models Math. Biosci 2007 300 3011

[26] CONAPO, http://www.conapo.gob.mx/work/models/CONAPO/Mapa_Ind_Dem18/index_2.html.

[27] I.G. Violaris, T. Lampros, K. Kalafatakis, G. Ntritsos, K. Kostikas, N. Giannakeas, M. Tsipouras, E. Glavas, D. Tsalikakis, A. Tzallas Modelling the COVID-19 pandemic: focusing on the case of Greece Epidemics 2023 100706

[28] F. Saldaña, H. Flores-Arguedas, A. Camacho, I. Barradas Modeling the transmission dynamics and the impact of the control interventions for the COVID-19 epidemic outbreak Math. Biosci. Eng. 2020 4165 4183

[29] B. Tang, X. Wang, Q. Li, N.L. Bragazzi, S. Tang, Y. Xiao, J. Wu Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions J. Clin. Med. 2020 32046137

[30] A. Acuña-Zegarra, M. Santana-Cibrian, J.X. Velasco-Hernández Modeling behavioral change and COVID-19 containment in México: a trade-of between lockdown and compliance Math. Biosci. 2020 108370

[31] E.A. Iboi, O. Sharomi, C.N. Ngonghala, A. Gumel Mathematical modeling and analysis of COVID-19 pandemic in Nigeria Math. Biosci. Eng. 2020 7192 7220

[32] Y. Liu, L.M. Yan, L. Wan, T.X. Xiang, A. Le, J.M. Liu, M. Peiris, L.L.M. Poon, W. Zhang Viral dynamics in mild and severe cases of COVID-19 Lancet 2020 656 657

[33] L. Zou, F. Ruan, M. Huang, L. Liang, H. Huang, Z. Hong, J. Yu, M. Kang, Y. Song, J. Xia, Q. Guo, T. Song, J. He, H.L. Yen, M. Peiris, J. Wu SARS-CoV-2 viral load in upper respiratory specimens of infected patients N. Engl. J. Med. 2020 1177 1179

[34] Y. Pan, D. Zhang, P. Yang, L.L.M. Poon, Q. Wang Viral load of SARS-CoV-2 in samples Lancet 2020 411 412

[35] E.A. Meyerowitz, A. Richterman, R.T. Gandhi, P.E. Sax Transmission of SARS-CoV-2: a review of viral, host, and environmental factors Ann. Intern. Med. 2020 69 79

[36] D. Kault Superspreaders, asymptomatics and COVID-19 elimination Med. J. Aust. 2021 140 140.e1

[37] B. Ridenhour, J.M. Kowalik, D.K. Shay Transmission of SARS-CoV-2: unraveling Ro : considerations for public health applications Am. J. Public Health 2014 e32 e41