R0: Determining the Spread of Disease

Epidemiology can calculate which diseases spread

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Sometimes a disease spreads like wildfire. Some don't. Some diseases just dies out. How do we figure out which diseases are going to spread and which won't?

Well, there's a term for this, R0, pronounced R "Nought".

Let's explain....

Imagine a class of 1st graders. One arrives to school sick. 

This kid's illness infects one child. The second child could then infect one child; the third child could infect another.

A chain of infections could spread the illness through the entire class.

In epidemiology , this is what we call a disease with an R0 of 1. Each case leads to one new case.

Definition

R0 is the basic reproductive number. It describes how many kids will get sick when one sick kid enters the classroom (the population) and all the kids are able to get sick (they are susceptible). It depends both on the disease itself and the interactions of the kids.

When R0's are higher than 1, more kids are infected. A high R0 isn't a more dangerous disease. A cold can have a high R0; a rare, but deadly, disease can have a low value, less than 1. 

Now back to the classroom.

R0 <1

Imagine that on average not every child infected another child. The first child infects a second; the second, a third. However, maybe the third doesn't infect any. 

​The illness would stop spreading.

This is what happens when R0 is less than 1.

Kids might get sick at first, but the disease will peter out.

R0 > 1

Let's go back to the first child, now imagine this child infects 2, those 2 kids infect 2 each (4 all together). In total, 7 would be infected. The last 4 could then infect 2 each, leading to 15 infections total. Pretty soon, there would be a lot of sick children.

This is what happens when R0 is 2 and no sick kid is kept home.

Susceptibility

In real life, not everyone is able to catch the bug. Some kids may be vaccinated. Some will get sick and can't get sick twice at the same time. Some kids will have gotten sick, recovered, and are immune. We say that not everyone is "susceptible".

In ongoing outbreaks the "Effective Reproductive Rate"(R) explains disease spread. This is the number of kids who get sick in a class that has a mix of kids who can't get sick and kids who can. (R increases with the proportion susceptible. R = R0x, where x is the fraction susceptible)

The number susceptible will change during an outbreak, as more kids get sick and recover or are vaccinated. Mixing of sick, immune, and recovered kids may also not be uniform.

Herd Immunity

If the first kid entered a room full of kids who were immune, disease would not spread. If almost every kid had been sick already and were immune, the disease wouldn't spread. If 8 out of 10 kids were vaccinated, the disease probably wouldn't spread.

The sick kid might not interact with the 2 out of 10 kids who could get sick.

We call this herd immunity. That is, the immunity of some kids protects other non-immune kids from getting sick. It is the fraction of kids who are immune such that each kid gets just one kid sick (on average). If one more kid were immune, the disease wouldn't spread.

If R0 is large, herd immunity protects only if many are immune. (Herd Immunity threshold = 1 - 1/R0) Larger the R0, the more kids need to be vaccinated.

Super-spreaders

Some kids spread more illness than others - like an ill teacher who works with every kid. Outbreaks can be more complicated than R0.

Real Life Examples

One of the most infectious diseases is Measles with an R0 between 12-18. Before measles vaccination, a child could infect 15 children in one classroom. Those 15 classmates could then each infect 15 schoolmates. Measles spreads fast. To avoid measles spreading, 83-95% need to be vaccinated.

We calculate R0 by observing how quickly the disease initially spreads. Ebola's n R0 is likely 1.34, 1.86 or 1.6-2.0. Smallpox was eradicated with an R0 of 3.5-6 which required 8 in 10 to be immune. Pertussis (Whooping Cough) has a high R0: 15-17. Untreated Tuberculosis' R0=10 while influenza's R0 is usually less than 2.

We can also also estimate R0 from contacts. In a classroom, contact might be kids playing blocks with hands, sneezes spreading infection. R0 value depends on this contact. It depends on how long illness lasts, how many contacts a kid has when ill, how often an illness spreads during each contact. 

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