# Factors of infection rate

I refer to J.H. Jones' Notes on R0.

## Preliminaries

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

$$ds/dt = -\beta s i$$

$$di/dt = \beta s i - \nu i$$

$$dr/dt = \nu i$$

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

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

$$R = \beta \cdot d_i$$

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 $$\tau_c$$, 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.

$$\beta = \tau_c \cdot c$$

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 $$\tau_h$$, i.e. the probability of getting infected when being in contact with an infected person for one hour

• mean duration of contact $$h_c$$ (measured in hours)

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

We then have

$$\beta = \tau_h \cdot h_c \cdot c$$

We can group factors:

$$\beta = \tau_c \cdot c$$

with $$\tau_c = \tau_h \cdot h_c$$, and

$$\beta = \tau_h \cdot h_d$$

with $$h_d = h_c \cdot c$$ the number of hours per day an individual is in contact with other persons.

## Question

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

I have not found texts relevant to your question and related to the spread of SARS-CoV-2. However, here is a publication that discusses the utility of contact duration data when constructing infection models, in the general case --

Simulation of an SEIR infectious disease model on the dynamic contact network of conference attendees

## Methods

At a conference, 405 attendees volunteered to wear RFID tags. Over two days, the time and duration of 28,540 face-to-face interactions between these individuals were recorded, with a temporal resolution of 20 seconds. Note the spatial limits of the RFID data:

RFID devices engage in bidirectional radio communication at multiple power levels, exchanging packets that contain a device-specific identifier. At low power level, packets can only be exchanged between tags within a radius of 1 to 2 meters.

From these data, three networks were constructed:

1. dynamic network (DYN) -- this network preserves the sequence and duration of contacts between individuals. Using the language of graph theory, this is a directed network with weighted edges.
So, if individual A comes into contact with individuals B and C, and individual B is infectious, individual A must contact individual B before individual C in order to have a chance of A → C transmission. Causality constrains the chain of transmission.
2. heterogenous network (HET) -- this network preserves the duration of contacts between individuals but disregards the sequence of contacts. The graph of this network is weighted but undirected. Adopting the previous example, B can infect C via A regardless of the relative order of the A ↔ B and A ↔ C interactions. Causality is suspended, and the probability of transmission depends on the number and duration of contacts.
3. homogenous network (HOM) -- this network inherits the topology of HET, but ignores the duration of contacts. The graph of this network is unweighted and undirected. Each contact event carries an equal probability of resulting in transmission, so the probability of transmission depends solely on the number of contacts.

To simulate the spread of an infectious disease over realistic time-scales, each network was longitudinally extended using three procedures:

1. repetition (REP) -- the recorded contacts are repeated with the same duration and order.
2. random shuffling (RAND-SH) -- the structure of the network is preserved, but the IDs associated with the nodes in the network are shuffled between iterations.
3. constrained shuffling (CONSTR-SH) -- same as RAND-SH, but the reordering of IDs is constrained to preserve individual social activity (i.e. IDs with many contacts are not assigned to a low-contact node in the network) and repeated contacts. This procedure rectifies the oversimplifications of REP and RAND-SH by acknowledging that people tend to interact with the same people day-to-day, with occasional contacts with strangers.

The authors applied a SEIR epidemic model (Susceptible, Exposed, Infectious, or Recovered). For each network-procedure pair, they considered two disease scenarios with different assumed values of mean latency period (σ -1), mean infectious period (ν -1), and transmission rate (β):

1. very short incubation and infectious periods
• σ -1 = 1 day
• ν -1 = 2 days
• β = 3.10 -4/s
2. short incubation and infectious periods
• σ -1 = 2 days
• ν -1 = 4 days
• β = 15.10 -5/s

A note from the authors on why these values were chosen:

These sets of parameter values were chosen to maintain the same value of β/ν, which is the biologic factor responsible for the rate of increase of cases during the epidemic outbreak, while changing the global timescales of incubation and infectious periods, and assessing the role played by the social factors embedded in the contact patterns.

## Results

Epidemic outbreaks in the three networks across the three data-extension procedures were compared by computing the mean R0, defined by the authors as the mean, over all iterations, of the number of secondary cases from the single initial randomly chosen infectious individual.

Considering the simplified REP procedure, higher values of R0 were observed in the HOM network compared to the HET and DYN networks, for all scenarios. The authors explain this observation by noting that the probability of rapid extinction of the pathogen was smallest for the HOM network; indeed, this held true independent of procedure.

In terms of the final number of cases (R) and the rate of spread, the HET and DYN networks were similar for both disease scenarios. The peak of the epidemic was reached first in the HOM network, on average, consistent with the reduced probability of extinction.

In the RAND-SH procedure, spread was slower but lasted longer, leading to a larger R for all networks. In general, greater shuffling of ID tags led to more efficient disease spread, such that R(RAND-SH) > R(CONSTR-SH) > R(REP) for all networks and scenarios. The interaction of more efficient disease spread in the RAND-SH procedure with increased values of σ -1 and ν -1 in the "short" scenario seems to produce consistently higher R0 values across networks (see Figure 3), though high variance in R0 means the effect of this interaction is not significant.

## Conclusions

By comparing the different network types across all procedures and scenarios, it was shown that ignoring the direction and duration of contacts overestimates both the rate at which a disease spreads and the total number of individuals infected. Additionally, shuffling individual positions in contact networks while preserving network topology increases disease spread when compared to a static model, though this effect is buffered by the "realistic" inclusion of preserved sub-networks of repeated contacts.

While this publication does not directly integrate factors for the duration and rate of contacts into the SEIR model, as suggested by Hans-Peter Stricker, the authors are able to show that mapping such a model to networks in which the direction and duration of contacts are encoded as edge weights leads to different measured outcomes (R0, R, time to peak) when compared to topologically identical networks lacking these encodings.

• Thanks for this great answer which will give me a lot of reading. It - and the study you point to - seems to be related to this question I asked at Mathematics SE: math.stackexchange.com/questions/3682077/… Jun 3, 2020 at 17:51
• @Hans-PeterStricker, if you do really believe mine is a "great answer", I'd appreciate if you accepted it. Given the bounty is now a week old and no other answers have popped up, I think it's appropriate. Jun 9, 2020 at 14:59
• I usually wait until the bounty would expire - which will be tomorrow - but thanks for the hint. Jun 9, 2020 at 17:13