The SIR dynamic model of infectious disease transmission and its analogy with chemical kinetics

The classic Susceptible-Infectious-Recovered (SIR) mathematical model of the dynamics of infectious disease transmission resembles a dynamic model of a batch reactor carrying out an autocatalytic reaction with catalyst deactivation. By making this analogy between disease transmission and chemical reactions, chemists and chemical engineers can peer into dynamic models of infectious disease transmission used to forecast epidemics and assess mitigation strategies. Mathematical models of the dynamics of infectious disease transmission [1, 2] are useful for fore-casting epidemics, assessing intervention strategies, and inferring properties of diseases. In compartmental epidemic models [3], each member of the population is categorized based on their disease status and (possibly) attributes. The dynamics of disease transmission are then typically modeled with a set of diﬀerential equations that describes the ﬂow of individuals to and from the compartments as the population mixes, the disease is spread/contracted, and infected individuals progress through the stages of the disease. Diﬀerential equations are a natural choice because we can make reasonable assumptions about the rates at which people are infected and recover. In this article, we highlight the analogy between compartmental epidemic models and dynamic models of chemical reactions.

Mathematical models of the dynamics of infectious disease transmission [ , ] are useful for forecasting epidemics, assessing intervention strategies, and inferring properties of diseases.
In compartmental epidemic models [ ], each member of the population is categorized based on their disease status and (possibly) attributes. The dynamics of disease transmission are then typically modeled with a set of differential equations that describes the flow of individuals to and from the compartments as the population mixes, the disease is spread/contracted, and infected individuals progress through the stages of the disease. Differential equations are a natural choice because we can make reasonable assumptions about the rates at which people are infected and recover. In this article, we highlight the analogy between compartmental epidemic models and dynamic models of chemical reactions. In the classic SIR model of an epidemic [ , -], each member of the population belongs to one of three compartments: Susceptible, Infectious, or Recovered. Fig. depicts the flow of individuals through compartments, assuming that the disease confers immunity to re-infection after recovery.
Susceptible folks can contract the disease if they come into contact with an infectious individual.
Once infected, they move into the infectious compartment, assuming zero delay between infection and the ability to transmit the disease. This is analogous to an irreversible autocatalytic chemical reaction [ , ] between a reactant, S, and catalyst, I: Infectious individuals eventually recover from the disease, entering the recovered compartment, and then cannot transmit the disease or contract it again. This is analogous to a reaction where the catalyst, I, irreversibly degrades or converts to a deactivated form R: So, the SIR model of an epidemic is analogous to an autocatalytic reaction with catalyst deactivation. ( ) The only two parameters in the SIR model are the transmission and recovery rate constants, α and γ, respectively. While γ could be estimated independently from studies on the duration of infectiousness, α could be identified by fitting differential eqns. In the SIR model, what happens if we introduce a single infectious individual into a population of entirely susceptible individuals? This is akin to introducing our deactivating auto-catalyst into a batch of reactant. Intuitively, if the catalyst has a sufficiently high activity and/or remains active long enough, it will initiate a reaction (an epidemic). To the contrary, if the catalyst has a low activity and/or quickly deactivates, it will not initiate a reaction. The activity and longevity of the catalyst are embedded in α and γ, respectively. See Appendix A. If R 0 < 1, the infectious recover quickly, the disease is not easily transmitted, and/or the mixing of susceptibles and infectious is not vigorous. Consequently, an epidemic will not ensue; [I](t) decreases monotonically. If R 0 > 1, the infectious are infectious for a long period of time, the disease is easily transmitted, and/or the mixing of susceptibles and infectious is vigorous. Consequently, the single infectious in- growth rate γ(R 0 −1) (see Appendix B). As the disease propagates, the concentration of susceptible individuals decreases, eventually causing the incidence rate of the disease to diminish. Interestingly, in conjunction with the infectious folks recovering, this causes the epidemic to die out before the entire population is infected. That is, an SIR model epidemic (case R 0 > 1) does not result in every susceptible member of the population being infected, even after an infinite amount of time (see Fig. ). From a chemical engineer's standpoint, the reaction begins to die out ( dt < 0) when the concentration of the reactant, [S], becomes so low that any given catalyst particle, I, is expected to deactivate before it can convert a reactant molecule, S, into another catalyst particle to replace itself.
i.e., an SIR epidemic dies out not because the population is depleted of susceptible folks, but rather because it is depleted of infectious folks [ , ]. The fraction of the reactant consumed (fraction of susceptibles infected) at the end of the reaction (epidemic) depends on both the activity and longevity of the catalyst (frequency of S-I contacts, transmissibility of the disease, time period of infectiousness; all of which are embedded in R 0 ). By the end of the epidemic, the fraction of the population that will not have been infected, The mechanism by which a vaccination program provides herd immunity to a population is mathematically similar to the time in Fig. when the epidemic begins to die off: owing to a sufficiently small concentration of susceptible folks (achieved by vaccination or, as is the case in Fig. , by a fraction of the population having been infected, recovered, and conferred immunity), the infectious recover faster than they spread the disease to the susceptible folks to replace themselves [ ]. A simple way to model vaccination in the SIR model is to allow flow in Fig. from the S category directly to the R category when a vaccine that confers complete immunity is administered to susceptible folks [ , , ]. A chemical engineering view of herd immunity is the reduction of the concentration of the reactant, [S], achieved by either vaccination or by recovery from infection, so that a catalyst particle, I, fed to the reactor is expected to deactivate before it encounters an S particle and auto-catalyzes a reaction S − − → I.  In conclusion, by making an analogy between disease transmission and chemical reactions, chemists and chemical engineers can peer into dynamic models of infectious disease transmission. Moreover, this is a nice example of how concepts in one field can aid understanding and generate insights in another field.

the appendix
For sections A, B, C, and D below, consider introducing one infectious individual into a population comprised of N − 1 susceptible folks. The initial conditions here are: A the basic reproduction number, R 0 dt | t=0 > 0, the disease will propagate, and an epidemic will ensue; if d[I] dt | t=0 < 0, the infectious individual will recover before he/she transmits the disease to others [ ]. According to eqn. and our initial conditions, under the approximation of large N, dt ) at t = 0 is equal to the incidence rate of the disease, i.e. the rate at which new infections occur, which is approximately α, minus the rate of recovery of the infectious individual, γ. Thus, if R 0 < 1, the infectious individual is expected to recover before they can transmit the disease to another person. If R 0 > 1, the infectious individual is expected to transmit the disease before they recover, thereby initiating an epidemic.

The incidence rate of the disease (number of new infections per time) according to eqn. is Nα[S][I].
Via the initial condition in eqn. and our approximation of eqn. as [S](t = 0) ≈ 1, the incidence rate of the disease, caused by this single infectious individual, is approximately α new infections per time over the course of their infectiousness. Since γ −1 is the mean infectious period, implied by Consequently, in the initial stage of the epidemic, we see exponential growth: is also a valid approximation for R 0 < 1, it reinforces that introducing an infectious individual into a completely susceptible population will not result in an epidemic if R 0 < 1, since [I](t) will show exponential decay if R 0 < 1.

C simulating the SIR model in Julia
which is equivalent to eqn. . [ ] Paul L Delamater, Erica J Street, Timothy F Leslie, Y Tony Yang, and Kathryn H Jacobsen. Complexity of the basic reproduction number (R 0 ). Emerging Infectious Diseases, ( ): , .