Basic reproduction number of epidemic models on sparse networks

Satoru Morita

Phys. Rev. E 106, 034318 – Published 22 September 2022

Abstract

The basic reproduction number $R_{0}$ is a standard indicator of infection control in epidemiology. Although $R_{0}$ has been studied extensively for deterministic epidemic models, it has not been formulated accurately for models adopting network structures. Here, we extend a four-compartment model that includes commonly used epidemic models to a Markov process on networks. By examining the Markov process in detail, we derive a canonical formula for $R_{0}$ involving two probability values. Numerical calculations show that the derived formula is a better approximation than the conventional formula when the network is very sparse. We propose this as a standard formula for controlling infections that can only be transmitted through intimate contact, where contacts between individuals can be described as a sparse network.

Received 30 December 2021
Revised 25 May 2022
Accepted 6 September 2022

DOI:https://doi.org/10.1103/PhysRevE.106.034318

Physics Subject Headings (PhySH)

Epidemic

Networks

Authors & Affiliations

Satoru Morita ^*

Department of Mathematical and Systems Engineering, Shizuoka University, Hamamatsu 432-8561, Japan

^*morita.satoru@shizuoka.ac.jp

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Vol. 106, Iss. 3 — September 2022

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Figure 1
Illustration of the calculation of probabilities $c_{1}$ and $c_{2}$ in terms of the rates of infection transmission $σ, γ$ , and $ω$ between individuals in susceptible (S), exposed (E), infectious (I), and recovered (R) states. Consider a pair of individuals $i, j$ who become I or E as a result of infection transmission between them. (a) shows the rates of transition between states E, I, R, and S. (b) shows the processes required for a newly exposed individual $i$ to recover and become susceptible again before the original infectious individual $j$ recovers. The probability of each process is shown on the right-hand side of the corresponding arrow. The product of the three probabilities gives probability $c_{1}$ in Eq. (12). (c) shows the processes required for the original infectious individual $i$ to recover and become susceptible again before a newly exposed individual $j$ recovers. There are multiple possible paths, and summing their probabilities yields probability $c_{2}$ in Eq. (16). The lightly shaded arrows indicate the process by which the I-S pair becomes I-E (i.e., infection occurs), and the probability that this occurs is given by Eq. (11) if there are no other infectious individuals adjacent to individual $i$ . In the main text, we calculate the average number of secondary infections from individual $j$ , indicated by the shaded circles.
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Figure 2
Comparison of theoretical and numerical calculations for the following networks: (a) two regular graphs with $k = 4$ (solid lines) and $k = 20$ (dotted lines); (b) random networks with 5000 nodes having degree 2 and 5000 nodes having degree 6, where the degree correlation is $r = 0.2$ (dotted lines), 0 (solid lines), and $- 0.2$ (dashed lines); (c) Barabási-Albert network with $〈 k 〉 = 4$ . We set $N = 10 000$ and $γ = 1$ . The graphs in the top row plot the theoretical calculated value of $R_{0}$ as a function of $β / γ$ . Here, the black curve corresponds to $R_{0} = k β / γ$ as obtained by Eq. (10). The purple, green, blue, and orange curves represent Eq. (21) or (20) for SIS, SEIS, SIRS 1, and SIRS 2, respectively, where the values of $c_{1}$ and $c_{2}$ are as listed in Table 2. The graphs in the middle row plot the relative frequency of infectious individuals ( $〈 I 〉 / N$ ) as a function of $β / γ$ by numerical simulation. The graphs in the bottom row plot the coefficient of variation $Std (I) / 〈 I 〉$ , the maximum of which corresponds to the epidemic threshold. Here, the long downward arrows represent the epidemic thresholds $β_{c}$ given by Eq. (22), and the short downward arrows represent those given by Eq. (25); these values are likewise shown in Table 2. Note that in the middle graph of (a), the results for SEIS and SIRS 1 with $k = 20$ nearly overlap.
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