Global convergence of COVID-19 basic reproduction number and estimation from early-time SIR dynamics

Gabriel G. Katul; Assaad Mrad; Sara Bonetti; Gabriele Manoli; Anthony J. Parolari

A heated dispute about the effectiveness versus risk of smallpox inoculation was playing 2 out in eighteenth-century France, which was to launch the use of mathematical models 3 this, many and indeed the principal problems of epidemiology on which 23 preventive measures largely depend, such as the rate of infection, the 24 frequency of outbreaks, and the loss of immunity, can scarcely ever be 25 resolved by any other methods than those of mathematical analysis. 26 The classic susceptible-infectious-recovered (SIR) paradigm, initiated in the late 27 1920s [6] , now provides a mathematical framework that describes the core transmission 28 dynamics of a range of human diseases [7] [8] [9] [10] [11] [12] , including COVID-19 [13] . A key 29 parameter in the SIR paradigm is the basic reproduction number (R o ). The R o is 30 defined by the average number of secondary cases arising from a typical primary case in 31 an entirely susceptible population of size S o [14] [15] [16] . The usefulness of R o and 32 uncertainty in its estimation are not a subject of debate, as reviewed elsewhere [17] , and 33 therefore are not further discussed here. 34 In the analysis herein, the SIR model is used to uncover generic spread mechanisms 35 observed by COVID-19 dynamics globally, especially in the early phases of infectious 36 spread. During this early period, potential controls were not effectively put in place or 37 enforced in many countries around the world despite early warning signals from China, 38 Iran, and later on, Italy. Hence, the early phases of COVID-19 spread in many countries 39 where controls were weak offer a unique perspective on the ensemble-behavior of 40 COVID-19 R o . The analysis shows that there is global convergence (i.e. across many 41 nations) to an uncontrolled R o = 4.5 for COVID-19 describing early times spread. The 42 implications for evaluating potential control strategies from this reference R o are briefly 43 discussed in the context of mortality and maximum infections. Mathematical models of disease spread assume that a population within a compartment 47 (e.g., city, region, country) can be subdivided into a set of distinct classes [11] . The SIR 48 model classifies individuals in the compartment as one of three classes: susceptible (S), 49 infectious (I), and recovered or removed (R). Infectious individuals spread the disease 50 to susceptible and remain in the infectious class for a given period of time known as the 51 April 14, 2020 2/15 infectious period before moving into the recovered (or removed) class. Individuals in the 52 recovered class are assumed to be immune for an extended period (or removed from the 53 population). For the total population N = S + I + R, the dynamical system describing 54 the SIR equations are given as assumption is that the number of deceased individuals is << N . The dynamical system 64 in equation (3) assumption in SIR dynamics is the use of the mass-action principle. As with all 74 compartment models, mass action assumes that the rate of encounter between I and S 75 is proportional to their product. For this assumption to hold, it requires that members 76 of I and S be uniformly distributed in the space of the compartment [18] . Individuals -77 unlike molecules in an ideal solution within a closed container -do not mix 78 homogeneously. At minimum, the use of the mass action principle serves as a reference 79 model to compare more detailed mechanisms or explore data.