10/30/22, 5:11 PM L9: SI Model: Network Science – CS-7280-O01
L9: SI Model
Figure 10.5 from Network Science by Albert-László Barabási
Suppose that we have individuals in the population and, according to the homogeneous mixing assumption, each individual has the same number of contacts (this is shown as
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In the SI model, there are two compartments: Susceptible (S) and Infected (I) individuals. To become infected, a susceptible individual must come in contact with an infected individual. If someone gets infected, they stay infected.
If and are the number of susceptible and infected individuals as functions of time, respectively, we have that . We typically normalize these two functions by the population size, and we work with the two population densities:
and , with .
The infection starts at time=0 with a single infected individual: .
Suppose that an S individual is in contact with only one infected individual. Let us define the parameter as follows: is the probability that S will become infected during an infinitesimal time interval of length .
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10/30/22, 5:11 PM L9: SI Model: Network Science – CS-7280-O01
Given that the S individual is in contact with infected individuals, this probability
increases to because the infection can take place independently through any of the infected contacts (the approximation is good as long as this probability is very small).
If the density of infected individuals is , then the probability that the S individual becomes infected is .
The infection process is always probabilistic but for simplicity, we can model it deterministically with a two-state continuous-time Markov process: an individual moves from the S to the I state with a transition rate .
So, if the density of S individuals is s(t), the increase in the density of infected individuals during is:
Thus the SI model can be described with the differential equation:
with initial condition .
This is a nonlinear differential equation (because of the quadratic term) but it can be solved noting that
where we replaced with for simplicity. Integrating both sides, we get that:
The initial condition gives us that this constant is equal to . So, if we exponentiate both sides of the previous equation we get that:
and so we get the closed-form solution for the density of infected individuals for :
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10/30/22, 5:11 PM L9: SI Model: Network Science – CS-7280-O01
This function is plotted at the visualization.
There are some important things to note about this function:
1. For small values of t, when the density of infected individuals is very small and the outbreak is only at its start, i(t) increases exponentially fast:
2. The time constant during that “exponential regime” is . This time constant is
often used to quantify how fast an outbreak spreads. This time constant decreases with both the infectiousness of the pathogen (quantified by ) and the number of contacts .
3. For large values of t, the density of infected individuals tends asymptotically to 1 – meaning that everyone gets infected.
Food For Thought
Perform the last derivation, showing how to get equation (1), in more detail.
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