Reproduction #s and Fatality Rates
Disclaimer: I am not an epidemiologist nor medical professional. This visualization is simple, may rely on bad assumptions and may unintentionally mislead. While I have taken pains to be cautious and evidence-based, it is always possible I am wrong in an important way. Nothing here should be taken as medical advice, authoritative or as reason to worry any more or any less.
Exponential Growth & Basic Reproduction
Infectious disease spread is characterized by the basic reproduction number (R0): the number of secondary cases expected to arise per case of infection within an entirely susceptible population and absent deliberate actions to reduce disease transmission. If R0 > 1 the virus can spread, whereas if R0 < 1 disease transmission will naturally die out.
R0 is a function of the following:
- Transmissibility: the probability a susceptible person will become infected given contact with an infected person. Pathogens can be transmitted via air (droplets and aerosol), physical contact and contaminated surfaces.
- Contact rate: the frequency that people experiencing infectiousness come into contact with those still susceptible.
- Duration of infectiousness: the average length of time an infected person can transmit the disease.
Estimates of Covid-19 R0 have been as low as 1.4 and as high as 6.47 (!). A comprehensive review of the literature thus far found a mean R0 value of 3.28. As a comparison, seasonal influenza has an R0 of ~1.3 and measles an R0 of ~15.
Fatality Rate
Fatality rate estimates tend to vary widely when analyzing epidemics. One reason for inflated estimates is described in the difference between Case Fatality Rate (CFR) and Infected Fatality Rate (IFR):
- IFR: fatalities per individuals infected. The World Health Organization estimated Covid-19 IFR at 0.3–1.0%.
- CFR: fatalities per cases reported, which omits asymptomatic and mild cases if testing is not widely available and thus may be misleadingly high. Covid-19 CFR estimates range from 1.6% to 3.4% to yet higher.
A good discussion of this “denominator problem” can be found here.
However, this is not the only cause of variation. Fatality rates is a lagging indicator due to the length of incubation (up to 14 days) and the time between symptom onset and death (2–8 weeks according to the WHO). An analysis that took this into account projects a significantly higher fatality rate.
John Ioannidis (the godfather of research reliability) has yet a different take, estimating a fatality rate of just 0.3% (CI 0.05% to 1%). He came to these numbers by “projecting the Diamond Princess mortality rate onto the age structure of the U.S. population” and emphasizes that with such a large uncertainty interval, we have little idea of what to expect.
Finally, there exists a particularly concerning and nonlinear influence on fatality rates: healthcare availability. If hospitals are overrun, severe cases will not get the care they need and thus be more likely to end in death.
Visualizing Varied Estimates
Varied R0 and fatality rate estimates yield decidedly different growth rates and population outcomes. Here we’ve illustrated epidemic behavior under more favorable (“Low”) and less favorable (“High”) such estimates using a simple R model that assumes no behavior change during the spread:
Effective Reproduction Rate & Flattening the Curve
Effective reproduction number (R) is the actual rate of transmission given public interventions and the changing proportion of susceptible individuals as the epidemic progresses.
Given Covid-19’s apparent high baseline transmissibility, we may not be able to force R<0. This means the infection will spread. However, we can flatten the curve to reduce the number of individuals simultaneously requiring medical care, which in turn may increase the quality and availability of care for those with severe cases.
If we look back to the R0 equation, we find three factors which, if reduced, would result in a lower effective reproduction rate (R):
Select research on the efficacy of such measures can be found here, here and here.
If a significant proportion of people take precautions, the rate of spread will slow and fewer people will get infected. Fatality rates will likely be lower because the medical system will be under less strain. Here is what some scenarios could look like (again assuming no behavior change over the course of the spread within this hypothetical community):
Important: Self-Isolation While Sick is Not a Panacea
While self-isolation is essential to reduce the spread of Covid-19, it is far from a complete solution: this pathogen appears to be spread by those who are pre-symptomatic and asymptomatic, not just by those with symptoms. Viral shedding often precede symptoms, as is illustrated here for the case of influenza:
Some research suggests that Covid-19 may be transmissible 1–2 days prior to symptom onset. Compounding the problem, initial symptoms have lower severity and thus may be ignored or explained as allergies or a mild cold, thus delaying self-isolation. Even worse: some of those infected will be entirely asymptomatic. One study estimates that with China, as many as 86% of transmissions were “undocumented” — i.e. from individuals with mild, limited, or no symptoms.
To reduce the population’s overall vulnerability, it is therefore critical to limit social contact and practice diligent hygiene 1) even when feeling perfectly healthy and 2) even while around others who appear perfectly healthy.
Appendix 1: Covid-19 vs other infectious diseases
For fun (fun?), here is Covid-19 compared to other infectious diseases based on reported unmitigated basic reproduction rate (R0) and CFR values.
Appendix 2: The Model
The gifs were created with a simple disease spread visualization model, written in R, that takes into account R0, CFR, serial interval and the final size relation.
It assumes a completely isolated, small community in which susceptible depletion kicks in, using the final size relation s∞ = S(0)e^[–R0(1–s∞)] [1 2]. Other assumptions include homogeneous contact & transmission rates between agents (whereas better models consider heterogeneous contact & transmission rates, e.g. [1, 2, 3]), that serial interval == generation interval and that no behavior change occurs over the course of the epidemic.
The code is here. Feel free to fork it; just keep in mind its limitations. Please let me know if you see any errors or have any concerns!
