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Related question at Biology SE: How to model social structure in SIR models


Preliminaries

Let me start with the definition of the reproduction number of an airborne virus, e.g. as given here, only slightly modified:

$$R_0 = \Big(\frac{\text{infection}}{\text{contact hour}}\Big)\cdot\Big(\frac{\text{contact hours}}{\text{day}}\Big)\cdot\Big(\frac{\text{days}}{\text{infection}}\Big) = \tau \cdot \overline{c} \cdot d$$

with

  • $\tau$ = the transmissibility (i.e., probability of infection given contact between a susceptible and infected individual)

  • $\overline{c}$ = the average rate of contact between individuals

  • $d$ = the duration of infectiousness

The underlying model is a population of $N$ persons being in ever changing physical contacts, assuming that there is a given probability $\tau/4$ of getting infected when being in contact with an infected person for at least say 15 minutes and at mean distance of say 1.5 meter.

Consider points in time separated by 15 minutes, i.e. $\{t_0, t_1, \dots, t_T\}$ with $t_{i+1} = t_i + dt$ with $dt$ = 15 minutes and $T \rightarrow \infty$.

At each given point in time $t$ we have a physical contact graph $c$ with $c_{ij} = 1$ when person $i$ is in physical contact (as defined above) with person $j$ and $c_{ij} = 0$ otherwise.

For physical reasons the maximal degree of this graph is say 10, e.g. when a person is standing in a dense crowd.

Consider a time series of physical contact graphs $\{ c(t_i)\}_{i \in \mathbb{N}}$, describing the (physical) social life of a population. Let $c_i(t) = \sum_{j\neq i} c_{ij}(t)$ be the number of persons that person $i$ is in contact with at time $t$.

For most individuals $i$ the number $c_i(t)=0$ most of the time, but the personal contact rate

$$\overline{c_i} = \frac{1}{T}\sum_{t=0}^T c_i(t) $$

may vary between different persons.

Consider the interpersonal contact rate $\overline{c_{ij}} = \frac{1}{T}\sum_{t=0}^T c_{ij}(t)$ with $\overline{c_i} = \sum_{j\neq i} \overline{c_{ij}}$.

Consider these definitions:

  • Person $i$ is close to person $j$ when $\overline{c_{ij}} > C_0$.

  • Person $i$ is acquainted with person $j$ when $C_0 > \overline{c_{ij}} > C_1$.

  • Person $i$ is a stranger to person $j$ when $C_1 > \overline{c_{ij}} > 0$.

with appropriately chosen fixed values $1 > C_0 > C_1 > 0$.

The physical social graph of the population can then be defined as the symmetric graph $\sigma$ with

$$\sigma_{ij} = \begin{cases} \displaystyle 0 \text{ when } \overline{c_{ij}} = 0 \\ \displaystyle 1 \text{ when $i$ is a stranger to $j$} \\ \displaystyle 2 \text{ when $i$ is acquainted with $j$} \\ \displaystyle 3 \text{ when $i$ is close to $j$} \\ \end{cases}$$

Now it comes to infection. The state of infection of a population is described by a vector $\{ \iota_i\}_{i \leq N}$ with $\iota_i = 1$ when person $i$ is infected and $\iota_i = 0$ otherwise. For the sake of simplicity let's assume that an individual becomes infectuous as soon as it gets infected (i.e. incubation time = 0), and that it recovers binarily, i.e. from one time step to the next.

A potential model of the disease consists of a time series of physical contact graphs $\{ c(t_i)\}_{i \in \mathbb{N}}$ and a time series of infection states $\{ \iota(t_i)\}_{i \in \mathbb{N}}$ that obey some restrictions.

To formulate these restrictions consider

  • the number of infected persons that person $i$ is in physical contact with at time $t$:

$$n_{i}(t) = \sum_{j\neq i}c_{ij}(t)\iota_j(t)$$

  • the rate of getting infected by simultaneous contact of person $i$ with $n$ individuals:

$$\alpha_i^{(n)}(t') = \frac{\big|\{t\leq t'\ |\ \iota_i(t) = 0 \textsf{ and } n_i(t) = n \textsf{ and } \iota_i(t+dt) = 1 \}\big|}{\big|\{t\leq t'\ |\ \iota_i(t) = 0 \textsf{ and } n_i(t) = n \}\big|}$$

  • the rate of recovering from the disease:

$$\rho_i(t') = \frac{\big|\{t\leq t'\ |\ \iota_i(t) = 1 \textsf{ and } \iota_i(t+dt) = 0 \}\big|}{\big|\{t\leq t'\ |\ \iota_i(t) = 1 \}\big|}$$

The (partly soft) restrictions now are:

  • $\lim_{t \rightarrow T} \alpha_i^{(n)}(t) = n \cdot \tau / 4$ for each person $i$ and each number $n$ of simultaneous contacts.

  • $\lim_{t \rightarrow T} \rho_i(t) = 1/\Delta =: \nu$ for each person $i$ with $\Delta = 96d$ and $d$ the duration of infectiousness (≈ infectedness) in days.

  • The maximal degree of the physical contact graph $c(t)$ is 10.

  • Contacts don't flip too frequently.

  • Contacts are sensibly distributed over persons and time.

For each model of the disease (as defined by these restrictions) one can just count the following numbers:

  • the number of recovered individuals $R(t) = \sum_{i=0}^N \iota_i(t-dt)(1- \iota_i(t))$

  • the number of infected individuals $I(t) = \sum_{i=0}^N \iota_i(t-dt)\iota_i(t)$

  • the number of susceptible individuals $S(t) = N - R(t) - I(t)$

Definition: A potential model of the disease is a SIR model, when there is a model-dependent constant $\overline{c}$ such that the numbers of susceptible, infected, and recovered individuals roughly evolve according to

$ds/dt = -\tau \overline{c} \cdot s i$

$di/dt = \tau \overline{c} \cdot s i - \nu \cdot i$

$dr/dt = \nu \cdot i$

with $\nu = 1/\Delta$ and normalized numbers $s = S/N$, $i = I/N$, $r = R/N$.

I assume that the last condition is automatically fulfilled (by definition).

Question

In the literature $\overline{c}$ is called and plays the role of an average rate of contact between individuals. But I assume when calculating such an average from the physical contact graph as

$$\overline{c} = \frac{1}{N} \sum_{i=0}^N \overline{c}_i$$

this would not do the job and generally fulfill the difference equations - because of too many characteristics of the contact and the social graph which would influence the proper number $\overline{c}$.

So my question is:

Is there a chance to find or define such graph characteristics - say $p_0, p_1, ..., p_M$ and a function $\textsf{c}$ such that

$$\overline{c} = \textsf{c}(p_0,\dots,p_M)$$

for almost all SIR models?

Candidates for such characteristics are:

  • average rate of contacts (see above)

  • standard deviation thereof

  • mean numbers of close, acquainted, and strange persons

  • conditional probabilities of being close to persons that are close to persons one is close to

  • mean frequency of change (volatility) of contacts

  • the frequency of big events (where many strangers come together for a considerable amout of time)


User @Dmitry suggested to "start with a simple model (based upon a small number of introduced characteristics) and try to expand it to address more complex dynamical properties".

Has anyone else already taken this approach?

References

These articles shed some interesting light on this question:

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    $\begingroup$ So, basically, your plan amounts to building a probabilistic, dynamical model of graph formation. I'm not aware of any general results in this direction. Perhaps you could start with a simple model (based upon a small number of introduced characteristics) and try to expand it to address more complex dynamical properties. $\endgroup$
    – Dmitry
    May 20 '20 at 18:48
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    $\begingroup$ I recommend you look at network centrality concepts such as eigenvector centrality. As you point out, we don't just care about the average number of contacts but who is contacting who. For example, it's bad if the contact graph is highly centralized, i.e. if there are a couple people who everyone contacts as those people will likely get sick meaning everyone is in contact with a few sick people. Network centrality concepts could lead to candidate $(p_0, p_1, ..., p_M)$. I agree with @Dmitry, work with something simpler such as a static model first. $\endgroup$ May 21 '20 at 18:58
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    $\begingroup$ amazon.com/Complex-Networks-Econometric-Society-Monographs/dp/… $\endgroup$
    – user762914
    May 25 '20 at 16:00
  • $\begingroup$ @Renard: Thank you so much for the hint! This book made my day. $\endgroup$ May 28 '20 at 6:15
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    $\begingroup$ @Dmitry: Thanks for having a look and your kind feedback. Once again I'll follow your advice. I'll keep you posted. $\endgroup$ Oct 10 '20 at 4:54
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Another related book is by Newman, Mark. Networks. Oxford university press, 2018. The SIR model is generally considered as the fully mixed model (the Erdos-Renyi random graph). Extensions to more general random graphs (the configuration model) can be found in that book.

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