Separable mixing: The general formulation and a particular example focusing on mask efficiency

M. C. J. Bootsma; K. M. D. Chan; O. Diekmann; H. Inaba; M. C. J. Bootsma; K. M. D. Chan; O. Diekmann; H. Inaba

doi:10.3934/mbe.2023785

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 10: 17661-17671. doi: 10.3934/mbe.2023785

Previous Article Next Article

Research article Special Issues

Separable mixing: The general formulation and a particular example focusing on mask efficiency

1.
Julius Centre for Health Sciences and Primary Care, University Medical Centre Utrecht, Utrecht University, Utrecht, The Netherlands
2.
Department of Mathematics, Faculty of Science, Utrecht University, Utrecht, The Netherlands
3.
Korteweg-de Vries Institute, University of Amsterdam, Amsterdam, The Netherlands
4.
Transtrend BV, Rotterdam, The Netherlands
5.
Faculty of Education, Tokyo Gakugei University, Koganei-shi, Tokyo, Japan

Received: 31 July 2023 Revised: 30 August 2023 Accepted: 05 September 2023 Published: 15 September 2023

The aim of this short note is twofold. First, we formulate the general Kermack-McKendrick epidemic model incorporating static heterogeneity and show how it simplifies to a scalar Renewal Equation (RE) when separable mixing is assumed. A key general feature is that all information about the heterogeneity is encoded in one nonlinear real valued function of a real variable. Next, we specialize the model ingredients so that we can study the efficiency of mask wearing as a non-pharmaceutical intervention to reduce the spread of an infectious disease. Our main result affirms that the best way to protect the population as a whole is to protect yourself. This qualitative insight was recently derived in the context of an SIR network model. Here, we extend the conclusion to proportionate mixing models incorporating a general function describing expected infectiousness as a function of time since infection.
- Kermack-McKendrick,
- epidemic model,
- heterogeneity,
- separable mixing,
- mask efficiency
Citation: M. C. J. Bootsma, K. M. D. Chan, O. Diekmann, H. Inaba. Separable mixing: The general formulation and a particular example focusing on mask efficiency[J]. Mathematical Biosciences and Engineering, 2023, 20(10): 17661-17671. doi: 10.3934/mbe.2023785

Related Papers:

Abstract

The aim of this short note is twofold. First, we formulate the general Kermack-McKendrick epidemic model incorporating static heterogeneity and show how it simplifies to a scalar Renewal Equation (RE) when separable mixing is assumed. A key general feature is that all information about the heterogeneity is encoded in one nonlinear real valued function of a real variable. Next, we specialize the model ingredients so that we can study the efficiency of mask wearing as a non-pharmaceutical intervention to reduce the spread of an infectious disease. Our main result affirms that the best way to protect the population as a whole is to protect yourself. This qualitative insight was recently derived in the context of an SIR network model. Here, we extend the conclusion to proportionate mixing models incorporating a general function describing expected infectiousness as a function of time since infection.

References

[1]	R. Pastor-Satorras, C. Castellano, The advantage of self-protecting interventions in mitigating epidemic circulation at the community level, Sci. Rep., 12 (2022). https://doi.org/10.1038/s41598-022-20152-4 doi: 10.1038/s41598-022-20152-4
[2]	M. C. J. Bootsma, K. M. D. Chan, O. Diekmann, H. Inaba, The effect of host population heterogeneity on epidemic outbreaks, (2023), submitted for publication.
[3]	O. Diekmann, J. A. P. Heesterbeek, T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton University Press, (2013).
[4]	W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London, 115 (1927), 700–721. http://doi.org/10.1098/rspa.1927.0118 doi: 10.1098/rspa.1927.0118
[5]	D. Breda, O. Diekmann, W. F. de Graaf, A. Pugliese, R. Vermiglio, On the formulation of epidemic models (an appraisal of Kermack and McKendrick), J. Biol. Dyn., 6 (2012), 103–117. https://doi.org/10.1080/17513758.2012.716454 doi: 10.1080/17513758.2012.716454
[6]	O. Diekmann, M. Gyllenberg, Abstract delay equations inspired by population dynamics, in Functional Analysis and Evolution Equations. The Günter Lumer Volume, Birkhauser Basel, 2007,187–200. https://doi.org/10.1007/978-3-7643-7794-6_12
[7]	O. Diekmann, P. Getto, M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2008), 1023–1069. https://doi.org/10.1137/060659211 doi: 10.1137/060659211
[8]	O. Diekmann, M. Gyllenberg, Equations with infinite delay: Blending the abstract and the concrete, J. Differ. Equ., 252 (2011), 819–851. https://doi.org/10.1016/j.jde.2011.09.038 doi: 10.1016/j.jde.2011.09.038
[9]	F. Brauer, C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, in Texts Appl. Math., Springer New York, 40 (2012). https://doi.org/10.1007/978-1-4614-1686-9
[10]	D. Breda, T. Kuniya, J. Ripoll, R. Vermiglio, Collocation of next-generation operators for computing the basic reproduction number of structured populations, J. Sci. Comput., 85 (2020). https://doi.org/10.1007/s10915-020-01339-1 doi: 10.1007/s10915-020-01339-1
[11]	D. Breda, F. Florian, J. Ripoll, R. Vermiglio, Efficient numerical computation of the basic reproduction number for structured populations, J. Comput. Appl. Math., 384 (2021). https://doi.org/10.1016/j.cam.2020.113165 doi: 10.1016/j.cam.2020.113165
[12]	O. Diekmann, H.G. Othmer, R. Planqué, M. C. J. Bootsma, The discrete-time Kermack-McKendrick model: A versatile and computationally attractive framework for modeling epidemics, PNAS, (2021). https://doi.org/10.1073/pnas.2106332118 doi: 10.1073/pnas.2106332118
[13]	E. Messina, M. Pezzella, A. Vecchio, Positive numerical approximation of an integro-differential epidemic model, Axioms, 11 (2022). https://doi.org/10.3390/axioms11020069 doi: 10.3390/axioms11020069
[14]	E. Messina, M. Pezzella, A. Vecchio, A non-standard numerical scheme for an age-of-infection epidemic model, J. Comput. Dyn., 9 (2022), 239–252. https://doi.org/10.3934/jcd.2021029 doi: 10.3934/jcd.2021029
[15]	E. Messina, M. Pezzella, A. Vecchio, Asymptotic solutions of non-linear implicit Volterra discrete equations, J. Comput. Appl. Math., 425 (2023). https://doi.org/10.1016/j.cam.2023.115068 doi: 10.1016/j.cam.2023.115068
[16]	O. Diekmann, H. Inaba, A systematic procedure for incorporating separable static heterogeneity into compartmental epidemic models, J. Math. Biol., 86 (2023). https://doi.org/10.1007/s00285-023-01865-0 doi: 10.1007/s00285-023-01865-0
[17]	O. Diekmann, J.A.P. Heesterbeek, J.A.J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382. https://doi.org/10.1007/BF00178324 doi: 10.1007/BF00178324
[18]	S.E. Eikenberry, M. Mancuso, E. Iboi, T. Phan, K. Eikenberry, Y. Kuang, et al., To mask or not to mask: Modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic, Infect. Dis. Model., 2020. https://doi.org/10.1016/j.idm.2020.04.001v doi: 10.1016/j.idm.2020.04.001v
[19]	C. N. Ngonghala, E. Iboi, S. Eikenberry, M. Scotch, C. R. MacIntyre, M. H. Bonds, et al., Mathematical assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel Coronavirus, Math. Biosci., 325 (2020). https://doi.org/10.1016/j.mbs.2020.108364 doi: 10.1016/j.mbs.2020.108364

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)