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Factors of infection rate

I refer to J.H. Jones’ Notes on R0.


The standard SIR model consists of a system of three differential equations

ds/dt = -βsi

di/dt = βsi – νi

dr/dt = νi

for the fractions s, i, r of susceptible, infected, and recovered (removed) individuals, dt = 1 day.

β is the infection rate, i.e. the number of individuals an infected indiviual infects per day, ν is the removal rate, being defined as the reciprocal of the duration of infectiousness di (measured in days), i.e. ν = 1/di. The reproduction number R is the product of infection rate and duration of infectiousness, i.e.

R = β · di

To reduce the growth of spreading of a disease, mitigation measures typically target at the infection rate, at least when a reduction of the duration of infectiousness is not at sight.

To estimate the impact of a mitigation measure on the infection rate, it is worth to consider it as the product of several factors, which may be influenced separately and more specifically.

One common approach is to consider the infection rate as the procuct of

  • transmissibility per contact τc, i.e. the probability of getting infected when being in one contact with an infected person and

  • mean rate of contact c, i.e. the number of single contacts an average individual has per day, i.e. β = τc · c.

Let alone that it is not so easy to define what a contact is, other factorizations are conceivable when considering these candidates:

  • transmissibility per hour τh, i.e. the probability of getting infected when being in contact with an infected person for one hour

  • mean duration of contact hc (measured in hours)

Let “being in contact” specifically mean “closer than 1.5 m in the average”.

We then have

β = τh · hc · c

We can group factors:

β = τc · c

with τc = τh · hc, and

β = τh · hd

with hd = hc · c the number of hours per day an individual is in contact with other persons.


I am looking for references where such factorizations of the infection rate are considered, especially in the context of Covid 19.

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