On Modeling of Interaction-Based Spread of Communicable Diseases

Abstract

We use a queuing model to study the spread of an infection due to interaction among individuals in a public facility. We provide tractable results for the probability that a susceptible individual leaves the facility infected. This model is then applied to study infection spread in a closed system like a large campus, community, and model the interaction among individuals in the multiple public facilities found in such systems. These public facilities could be restaurants, shopping malls, public transportation, etc. We study the impact of relative timescales of the Close Contact Time (CCT) and the individuals’ stay time in a facility on the spread of the virus. The key contribution is on using queuing theory to model time-spread of an infection in a closed population.

A Expressions for \(\varPsi (x,u)\) and \(\varGamma (x,u)\) of Sect. 4

We will first find $\varPsi (x,u)$. Since we are given that the system is Idle at time 0 and using the memoryless property of the exponential distribution, we can claim that the susceptible individual sees an alternating renewal process $I_1$, $B_1$, $I_2$, $B_2$, $I_3$, $B_3$, $\ldots$ where $I_i$ (resp. $B_i$ is the random variable corresponding to the $i^{th}$ Idle period (resp. Busy period). $I_i$ are independent and distributed as $Exp(\lambda )$, while $B_i$ have distribution of busy period of M/M/$\infty$ queue with arrival rate $\lambda$ and service requirement $Exp(\mu )$. Let $\varPsi _0(x,u) = P(\mathrm {Total\ Idle\ Period\ length} > u | \mathrm {Starting\ with\ idle\ period})$ so that $\varPsi (x,u) = 1- \varPsi _0(x,x-u)$. Using [16, Theorem 2.1], we can show that

$$\begin{aligned} \varPsi _0(x,u)= & {} \sum _{n=0}^{\infty } (E^{(n)} (x-u) - E^{(n+1)}(x-u)) B^{(n)}(u), \end{aligned}$$

where $E^{(n)}(\cdot )$ (resp. $B^{(n)}(\cdot )$ is $n-$fold convolution of $Exp(\lambda )$ (resp., $B(\cdot )$).

The distribution $\varGamma (x,u)$ is obtained again using [16, Theorem 2.1]

$$\begin{aligned} 1-\varGamma (x,u)= & {} \sum _{n=0}^{\infty } (C_n (u) -C_{n+1}(u)) E^{(n)}(x-u), \end{aligned}$$

where $C_n(\cdot ) = {\tilde{B}}*B^{(n-1)}(\cdot )$ for $n\ge 1$, i.e., convolution of ${\tilde{B}}(\cdot )$ and $B^{(n-1)}(\cdot )$, with $C_0(u)=1,\ \forall u$.