Global convergence of COVID-19 basic reproduction number and estimation from early-time SIR dynamics - PubMed (original) (raw)

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

Gabriel G Katul et al. PLoS One. 2020.

Abstract

The SIR ('susceptible-infectious-recovered') formulation is used to uncover the generic spread mechanisms observed by COVID-19 dynamics globally, especially in the early phases of infectious spread. During this early period, potential controls were not effectively put in place or enforced in many countries. Hence, the early phases of COVID-19 spread in countries where controls were weak offer a unique perspective on the ensemble-behavior of COVID-19 basic reproduction number Ro inferred from SIR formulation. The work here shows that there is global convergence (i.e., across many nations) to an uncontrolled Ro = 4.5 that describes the early time spread of COVID-19. This value is in agreement with independent estimates from other sources reviewed here and adds to the growing consensus that the early estimate of Ro = 2.2 adopted by the World Health Organization is low. A reconciliation between power-law and exponential growth predictions is also featured within the confines of the SIR formulation. The effects of testing ramp-up and the role of 'super-spreaders' on the inference of Ro are analyzed using idealized scenarios. Implications for evaluating potential control strategies from this uncontrolled Ro are briefly discussed in the context of the maximum possible infected fraction of the population (needed to assess health care capacity) and mortality (especially in the USA given diverging projections). Model results indicate that if intervention measures still result in Ro > 2.7 within 44 days after first infection, intervention is unlikely to be effective in general for COVID-19.

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Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Fig 1

Fig 1. Timeline of the COVID-19 R o estimates.

Symbols represent studies listed in Appendix (Table 1) while the red dashed line marks R o = 4.5 derived from this study. An initial R o = 2.2 was initially adopted by the World Health Organization (WHO).

Fig 2

Fig 2. Phase space and temporal trends of the SIR model.

(a) S(t), I(t), R(t) normalized by S o as a function of dimensionless time t* = γt with S o = 100, 000, γ = (1/14) d−1, and R o = 4.5. (b) i = I/S o in dimensionless time t* = γt for early times t* < 1 revealing strictly exponential growth (dashed) and deviations from exponential (SIR solution). (c) di/dt* with i in linear and (d) double-log representations. The dashed line is (R o − 1) where R o = 4.5. Declines from the dashed line reflect the incipient point where i(t) deviates appreciably from exponential growth. Note how the early-time slope (R o − 1) is emphasized in the double-log representation.

Fig 3

Fig 3. Comparison between di/dt and i for 57 countries.

The dashed line is (R o − 1)γ, where R o = 4.5, and γ = (1/14)_d_−1. Negative deviations from the dashed line reflect deviations from exponential in this phase-space representation.

Fig 4

Fig 4. Same as Fig 3 but for sample countries.

(a) the United States of America (US), the United Kingdom (UK), and Canada (CA); (b) Italy (IT), Spain (ES), and France (FR); (c) Belgium (BE), Germany (DE), the Netherlands (NL); (d) Australia (AU), New Zealand (NZ), and South Africa (ZA).

Fig 5

Fig 5. Same as Fig 3 but for sample UTLAs in the UK (a) and provinces in Italy (c).

Selected UTLAs and provinces are shown in panels b and d, respectively.

Fig 6

Fig 6. Modeled di/dt* as a function of i when considering a dynamic R o,d.

Five scenarios are illustrated (inset): no intervention (red) with R o = 4.5 set to its uncontrolled value, R o,c = 1.1 (epidemic near containment) and k c = 0.7 (blue), R o,c = 1.1 and k c = 0.15 (magenta), R o,c = 2.5 (typical of countries with strong initial intervention) and k c = 0.7 (green). The other parameters of the logistic functions are R o,u = 4.5 and _t_50 = 1.5/γ.

Fig 7

Fig 7. SIR model with super-spreaders.

Modeled I(x, y, t) at selected t* = γt showing the progression of disease outbreak in space due to mobility of super-spreaders only.

Fig 8

Fig 8. SIR model with super-spreaders: spatially integrated results.

(a) Modeled < _s_(_t_) > and < _i_(_t_) >, where < . > is spatially integrated quantities (the inset shows the Gaussian spatial spread kernel). (b) The early times dynamics showing the effects of super-spreaders in the phase of d < _i_ >/dt versus < _i_ >. The line (R o − 1)γ with R o = 4.5 is shown for reference. This effect is similar to those reported in the UK and Italy regions.

Fig 9

Fig 9. Relation between maximum infection fraction i max = I max/S o and R o.

Fig 10

Fig 10. Relation between total infection fraction 1 − S(∞)/S o and R o.

Fig 11

Fig 11. Relation between total mortality (M o) and R o for different values of α m and assuming S o = 327M.

Fig 12

Fig 12. Variation in cumulative number of infected relative to S o (top) and in maximum number of infected relative to S o (bottom).

The logistic form of R o was used (Eq (9)). The R o was set to vary from R o,c = 4.5 to R o,u = 1.0. The t*,50 and t*,80 are the dimensionless times at which R o is half and 80% through the the total decline.

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