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Impact of the infectious period on epidemics
Robert R Wilkinson1,2, Kieran J Sharkey2
1Department of Applied Mathematics, Liverpool John Moores University, Byrom Street, Liverpool L3 5UX, England, United Kingdom.
Abstract:
The duration of the infectious period is a crucial determinant of the ability of an infectious disease to spread. We consider an epidemic model that is network based and non-Markovian, containing classic Kermack-McKendrick, pairwise, message passing, and spatial models as special cases. For this model, we prove a monotonic relationship between the variability of the infectious period (with fixed mean) and the probability that the infection will reach any given subset of the population by any given time. For certain families of distributions, this result implies that epidemic severity is decreasing with respect to the variance of the infectious period. The striking importance of this relationship is demonstrated numerically. We then prove, with a fixed basic reproductive ratio (R_{0}), a monotonic relationship between the variability of the posterior transmission probability (which is a function of the infectious period) and the probability that the infection will reach any given subset of the population by any given time. Thus again, even when R_{0} is fixed, variability of the infectious period tends to dampen the epidemic. Numerical results illustrate this but indicate the relationship is weaker. We then show how our results apply to message passing, pairwise, and Kermack-McKendrick epidemic models, even when they are not exactly consistent with the stochastic dynamics. For Poissonian contact processes, and arbitrarily distributed infectious periods, we demonstrate how systems of delay differential equations and ordinary differential equations can provide upper and lower bounds, respectively, for the probability that any given individual has been infected by any given time.
Insights
Increased variability in infectious period duration, even with a fixed average, reduces epidemic spread. This finding holds across various epidemic models, suggesting that longer, more varied infectious periods can dampen disease transmission.
Area of Science:
- Epidemiology
- Mathematical Biology
- Network Science
Background:
- The duration of infectiousness is a critical factor influencing disease transmission dynamics.
- Existing epidemic models often assume fixed or exponentially distributed infectious periods.
Purpose of the Study:
- To investigate the impact of infectious period variability on epidemic spread in a non-Markovian network-based model.
- To establish relationships between infectious period variance and epidemic severity, independent of the mean and basic reproductive ratio (R0).
Main Methods:
- Developed a general, non-Markovian, network-based epidemic model encompassing Kermack-McKendrick, pairwise, and message passing models.
- Proved monotonic relationships between infectious period variability and epidemic spread probability.
- Utilized numerical simulations to demonstrate the impact of variance on epidemic severity.
- Derived bounds for infection probability using delay differential equations and ordinary differential equations.
Main Results:
- A monotonic relationship exists between infectious period variability (fixed mean) and the probability of infection reaching any population subset by a given time.
- Increased variance in the infectious period tends to decrease epidemic severity.
- Even with a fixed basic reproductive ratio (R0), higher variability in infectious period dampens epidemic spread.
- Delay differential equations and ordinary differential equations provide bounds for infection probability.
Conclusions:
- Variability in infectious period duration is a significant factor in controlling epidemic spread, independent of average duration or R0.
- The findings have broad applicability to various epidemic models, including those not strictly adhering to stochastic dynamics.
- Mathematical models, including ODEs and DDEs, can effectively bound epidemic outcomes based on infectious period characteristics.
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