The Mathematics of a Pandemic

[Avalux]

I was reading this last night and want to share with everyone here. It's not meant to scare anyone but it will give you an idea of how an infection that's easily transmittable can spread over a short period of time, just from one person.

We first heard reports of swine flu on Thursday of last week, so if we go by that measure, today is Day 6. The CDC and WHO agree the spread of swine flu cannot be contained; from reports today, it seems to be all over the world now. I am not predicting the current swine flu outbreak will follow the below extrapolation but it is interesting to see the math.

This flu-wiki diary was created in 2005 as a "what if" for an avian flu pandemic:

Pandemic Velocity Calculations

I've put together a spreadsheet, but am unable to get it into a form that will work here, so
the two variables are:
1. Average number of people spread to "R"
2. Average number of days to spread to average number of people "D"

And Patient Zero - First h2h patient who lives work in a major urban/metro setting worldwide

The results are interesting...

with R=5 and D=2
Day #infected
0 1
2 5
4 25
6 125
8 625
10 3,125
12 15,625
14 78,125
16 390,625
18 1,953,125
20 9,765,625
22 48,828,125
24 244,140,625
26 1,220,703,125

http://www.newfluwiki2.com/showDiary.do?diaryId=546

[mike_c]

1. does your model take individual time courses into account...?

To clarify: each infected person is themselves infectious to others for a finite period of time, after which they either die or recover, but in any event are no longer infectious. That's why a strictly exponential growth model is not best. I'd use something like:

I_t+1 = I_t + asI_tU_t - rI_t - mI_t

I_t = infected persons at time t
U_t = uninfected persons at time t
a = encounter rate with uninfected persons
s = susceptibility rate (proportion of uninfected persons infected during each encounter with a carrier)
r = recovery rate
m = mortality rate

Lots of good models out there-- this one's off the top of my head.

[readmoreoften]

2. That's a really good point. Maybe you should add it to the wiki?

I'm glad that people are discussing this in terms of mathematical models because I think that's really what it comes down to. It doesn't seem like this virus is incredibly infectious so far, given the data.

Thanks for posting.

[Avalux]

3. I'm not a mathemetician - you obviously know a lot more about this than I do.

I just thought the post and the subsequent thread was interesting. Maybe you can help me understand your model better and please - add it to the wiki thread too. :hi:

[mike_c]

9. it really was off the top of my head...

...but here's some more explanation.

I used a discrete rather than continuous model just because it was easier to think in discrete time steps rather than differential equations. Some folks around here will object. I agree in advance.

So, the first term I_t is the number of people currently infected-- this is the pool of folks who currently have the disease and can infect others. I'm assuming they have some average infectiousness throughout the time they're infectious.

Add to this the number of new infections: asI_tU_t where I_t is the number of currently infected people spreading the disease, U_t is the number of people who are uninfected and susceptible. This latter term is necessary because you can only be infected once-- if you recover, you're immune. Otherwise, I'd just consider the number of susceptible people to be effectively infinite and omit the term. Not every infected person will encounter an uninfected person, so a is the encounter rate, and not every encounter will lead to a new infection, so s is the susceptibility rate (think of it as the transmission frequency, and I should have used t to denote it, but I used t for time.) :-)

So the number of infected people in the next time step is I_t + asI_tU_t. But some of those people recover during each time step, so we subtract the number who recover rI_t, and some die, so we subtract them, too: - mI_t.

Real epidemiologists will find lots to criticize-- this is just a crude mass action model, but it's a good start, I think.

[dos pelos]

7. It's all about the variable "a"

It is all about the encounter rate.Virulence is environmentally determined by " a".In an environment of easy encounter,easy transmission,the most virulent forms of a virus proliferate,the less virulent forms are out competed.If "a" is a lesser value,if the environment is marked by slow transmission,virulent forms die out in their host(which also dies),untransmitted.In such an environment the less virulent forms dominate.Environment determines,selects for virulence.Variable "a" is everything.This is why,in crowded,unsanitary Mexico City,easy transmission due to high encounter rates selects for the proliferation of the most virulent forms.This is why,also,this flu virus will be much less virulent in the US.

[mike_c]

11. but...

Regarding "This is why,also,this flu virus will be much less virulent in the US."

That assumes selection favoring the less virulent forms in low transmission environments, which could take some time to appear. I'd be surprised if that's the full explanation of lower APPARENT virulence in the U.S. at more-or-less the same time the patients infected immediately beforehand in Mexico were dying faster.

How long would you expect the shift to lower virulence to take to become significant?

[TheCoxwain]

4. I can think of another Major flaw in your simple model.

You assume that every person will infect 5 other uninfected people..


This might be true during the initial phase - but as the disease spreads - you will have the same people running into each other. This coupled with immunity of a certain portion of the population - the disease will smooth of logarithmically rather than a increase in a geometric fashion as you have postulated.

In reality - ( My guess) the growth will resemble a "S" curve during growth and an exponential decline during the regression of the disease.

[hvn_nbr_2]

5. You mean the Ponzi model of epidemiology won't work?

If the "flu wiki" had continued their calculations for two more cycles, it would show that all 25 billion earthlings had become infected.

[Avalux]

8. OMG you're right!

Now I'm a little embarrassed I posted it (didn't write it myself).

[mike_c]

12. and in another interation or three...

...all the matter in the universe would be composed of swine flu virons!

[nadinbrzezinski]

6. They are more like Bell curves

but yeah I get your point

:-)

And I am not a mathematics nerd either

Why I have been following numbers

[TheCoxwain]

10. I guess you are right .. Bell curve makes more sense.