A frailty model for intervention effectiveness against disease transmission when implemented with unobservable heterogeneity

Ping Yan; Ping Yan

doi:10.3934/mbe.2018012

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 1: 275-298. doi: 10.3934/mbe.2018012

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A frailty model for intervention effectiveness against disease transmission when implemented with unobservable heterogeneity

Ping Yan

Infectious Diseases Prevention and Control Branch, Public Health Agency of Canada, Ottawa, Ontario, Canada, K1A 0K9

Received: 02 July 2016 Revised: 01 March 2017 Published: 01 February 2018
MSC : Primary: 92B05, 60E10, 60E15

For an intervention against the spread of communicable diseases, the idealized situation is when individuals fully comply with the intervention and the exposure to the infectious agent is comparable across all individuals. Some level of non-compliance is likely where the intervention is widely implemented. The focus is on a more accurate view of its effects population-wide. A frailty model is applied. Qualitative analysis, in mathematical terms, reveals how large variability in compliance renders the intervention less effective. This finding sharpens our vague, intuitive and empirical notions. An effective reproduction number in the presence of frailty is defined and is shown to be invariant with respect to the time-scale of disease progression. This makes the results in this paper valid for a wide spectrum of acute and chronic infectious diseases. Quantitative analysis by comparing numerical results shows that they are also robust with respect to assumptions on disease progression structure and distributions, such as with or without the latent period and the assumed distributions of latent and infectious periods.
- Frailty model,
- heterogeneity,
- infectious diseases control,
- intervention,
- reproduction number
Citation: Ping Yan. A frailty model for intervention effectiveness against disease transmission when implemented with unobservable heterogeneity[J]. Mathematical Biosciences and Engineering, 2018, 15(1): 275-298. doi: 10.3934/mbe.2018012

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Abstract

For an intervention against the spread of communicable diseases, the idealized situation is when individuals fully comply with the intervention and the exposure to the infectious agent is comparable across all individuals. Some level of non-compliance is likely where the intervention is widely implemented. The focus is on a more accurate view of its effects population-wide. A frailty model is applied. Qualitative analysis, in mathematical terms, reveals how large variability in compliance renders the intervention less effective. This finding sharpens our vague, intuitive and empirical notions. An effective reproduction number in the presence of frailty is defined and is shown to be invariant with respect to the time-scale of disease progression. This makes the results in this paper valid for a wide spectrum of acute and chronic infectious diseases. Quantitative analysis by comparing numerical results shows that they are also robust with respect to assumptions on disease progression structure and distributions, such as with or without the latent period and the assumed distributions of latent and infectious periods.

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