Interpreting Logistic Equation in Foot and Mouth Disease Epidemic

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Purpose

This page offers a qualitative interpretation for the logistic equation's good fit to the Foot and Mouth Disease epidemic that occurred in the United Kingdom in 2001. (See here for more details).

The interpretation develops a connection between the logistic equation and the more sophisticated epidemiological models used by Howard [HOW011100], Ferguson, and the UK Chief Scientist and the Ministry of Agriculture, Fisheries and Food (MAFF), in fighting the 2001 epidemic.

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Epidemiological Definitions

From [HOW011100]

	Term	Definition
	Latent Period	The interval between the point of infection and the onset of infectiousness
(2)	Asymptomatic infectious period	The interval between the onset of infectiousness and the appearance of clinical signs of disease
(3)	Incubation period (sum of (1) and (2)	The interval between the point of infection and the appearance of clinical signs of disease
(4)	Force of Infection	The rate at which susceptible herds acquire infection from infectious herds
(5)	Basic Reproduction Number	The average number of secondary herds infected by the introduction of one primary infected herd into a susceptible population.

"The value of the basic reproduction number at any time is given by the length of the infectious period multiplied by the the force of infection at that time"
[HOW011100,p3]

So

BRN(t) = Infectious Period x Force of Infection(t) 	...(1)

"For an epidemic to be sustained, the basic reproduction number has to be greater than one"

ie.

BRN(t) > 1 				

Now,

"The rate of spread of infection between herds, by whatever mechanism, is proportional to the number of susceptible herds multiplied by the density of infectious herds at that time." - [HOW011100,p191] ...(2)

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Developing the Logistic Equation

Assume that the susceptible herds and density of infected herds are both functions of time, then [HOW011100]'s definition (2) can be written:

	
d(Infected Herds)/dt = Ka  x Susceptible Herds(t) x Density of Infected Herds(t)
						... (3)
						
Let 	P = infected herds
	S = susceptible herds
	Ka = constant of proportionality

What does "density of infected herds" mean ? In a highly controlled experimental situation, you might have enough data to be able to represent the spatial distribution of herds. But that's a heck of a lot of work, and if the data you have to use in real life is dodgy, you might be better to work on aggregates or averages.

Let's make a simplification: an infected herd means one capable of passing on an infection. Strictly, this is ignoring the latent period between the time when an animal in a herd acquires an infection and when it is capable of passing it on. This simplification would be valid if that doesn't matter too much, or can't be reliably measured.

Assume that:

Density of Infected Herds(t) = ka / Ka  [P(t) - E(t)] 	 ...(4)

where E(t) = total  infected herds completely exterminated 
		(incapable of infecting,  by any means) 

dP/dt = ka S(t) [ P(t) - E(t) ] 		...(5)

For the epidemic to end, we need the population of confirmed cases (infected herds) to reach a limit - with no further change. The objective function is then:

dP/dt = 0					...(6)

We don't know what sort of function to use for dP/dt. Alfred Lotka postulated that it might be some function of the population size:

dP/dt = f(P) = 0				...(7)

The Taylor series expansion of any function f(P) gives:

dP/dt = bP + dP2 + gP3 + .... 			...(8)

Using one term of the taylor series expansion for f(P) gives an exponentially increasing population - and can't satisfy the objective (6).

Taking the first two terms of the taylor series expansion is the simplest approximation to f(P) that will permit a non-trivial solution to (6).

dP/dt = bP + dP2 					...(9)

Solving (6) and (9) gives

 0	= Plimit ( b + dPlimit )
 

hence

b + dPlimit= 0
 
 Plimit = - b /d			...(9a)

( 
Since 

 Plimit ≥ 0
 
 we would expect that b  ≥ 0 and d ≤ 0  in equation (9).
 )
 

Now what is a susceptible herd (S(t))? We suspect there is some limit to the number of infected herds - let's forget for the moment that we've just postulated an expression for it in equation (9a).

Let's suppose that anything beyond the as yet unknown limit is not considered susceptible (so, for example, herds in Australia would not be included as susceptible when we were studying an outbreak in the United Kingdom).

Let 

  klimit = P(t=¥)

then:

S(t) = klimit - P(t)				 ...(10)

where klimit = a positive constant > 0

Now substitute (10) into (5):

dP/dt = ka [ klimit - P(t) ]  [ P(t) - E(t) ] 	...(11)

dP/dt = ka [ klimit P(t) - P(t)2 + P(t) E(t) - klimit E(t) ]

then

let  	b = ka klimit

	d = ka

dP/dt = b P(t) - d P(t)2 +d P(t) E(t) - b E(t)	...(12)

If you exterminated no infected herds, or were so slow in doing it that the virus was free to spread in a way that made no difference compared to zero extermination (for example, by the time you culled, all the infectivity had gone), then E(t) = 0

At the time of writing there is no clear evidence as to how much infection occurs after reporting, case confirmation, or culling (see especially Ferguson's difficulty with estimating his r_I factor). It's then reasonable to explore the possibility:

E(t) = 0 				...(13)

then substituting in (12):

dP/dt = b P(t) - d P(t)2 		...(14)

which is just the logistic equation and dP/dt = 0 means end of the epidemic.

The goodness of fit of such a logistic equation to the data is evidence that the suppositions are correct: namely the rate of spread of infection is determined primarily by what happens before an infected herd ceases to exist (I'm being careful here to allow that carcasses might still exist from an infection standpoint); and closeness to an infected doesn't particularly matter as far as population aggregates are concerned. This is simplistic, but if reality is like that - e.g. wind or other transport mechanisms are more important than assumed to date - then the logistic fit will be good, or at least better than alternative models.

But there might be a better qualitative explanation.

The number of exterminated cases will be proportional to the infected cases, but with perhaps a loss factor and some lag due to the lead time of detecting, then organising slaughter and disposal:

E(t) = kvulnerable   P(t-td) 			...(15)

where 0 <= kvulnerable < 1 

k_vulnerable is a loss factor that represents the proportion of the infected population that will be vulernable to treatment - i.e. detected, and culled. Surveillance and detection will never be 100% (e.g. asymptomatic carriers), and some detected farms will manage to infect others after slaughtering (sloppy infection control by slaughtermen etc.,.).

t_d is the time lag between diagnosis and slaughter & disposal,

and (15) means you will are trying to exterminate all the cases infected t_d days previous.

but if detection and slaughter is prompt ( as pre-emptive neighbour culling might cause) then, t_d=0

E(t) = kvulnerable * P(t) 		...(16)

This is saying you are exterminating a high proportion of the infected cases on the day they get infected. It's more than saying a high proportion of confirmed cases is exterminated.

More generally, you could formulate the problem as a pair of differential equations relating E(t) and P(t). And then, you could think of more complicated ways to define the terms, and bring in more variables and more equations. So you might represent the distinction between infected and confirmed cases, as Ferguson et al did. Then you could go on to add even more variables than Ferguson (e.g. for species differences). Where do you stop ? At some point, there won't be sufficient data to make any meaningful correlation with the data. Ferguson's inability to estimate r_I suggests he went too far. The suggestion here is that you stop when the model you get fits good enough, and that (16) might be good enough.

substituting in (11):

     dP/dt 	= ka  [ klimit - P(t) ]  [ P(t) - kvulnerable   P(t) ] 	...(17)

          	= ka  [ klimit - P(t)]  P(t)  [ 1 - kvulnerable ] 
	
         	= ka  [1-kvulnerable]  klimit  P(t) - ka  [1-kvulnerable]  P(t)2

so
	dP/dt 	= b'  P(t) - d'  P(t)2

which is the logistic equation

where

	d' = ka  [1-kvulnerable]

	b' = ka   [1-kvulnerable]   klimit

    	   = d' x klimit

with  b'/d' = klimit

Note b'/d' is just the expression for the asymptote for P(t) (from the solution to the differential equation), and hence matches the interpretation put on k_limit.

In using MAFF web-published statistics, we have only confirmed cases data to go by. Provided the confirmed cases are proportional to the infected cases, the analysis still holds.

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Conclusions

If the size of the exterminated infected population E(t) is either effectively zero , or proportional to the size of the infected population, you get a logistic equation.

Anecdotal evidence suggests that delays to slaughter were significant in the 2001 UK outbreak. But it's not clear whether the pre-emptive nature of the 3 km cull might still have left an effective t_d of zero.

To decide conclusively which of the two explanations is correct, you would need more data and research that was able to discriminate what amount of infection occurred after case confirmation, and quality evidence of the slaughter delays.

In the context of the UK FMD epidemic, the pre-emptive nature of 3 km culling may have helped exterminate FMD in the quicker manner predicted by the logistic equation. Execution failures with the pre-emptive cull may have led to FMD lasting longer than the logistic equation projections suggested.

Further research, assuming data can be obtained, might help decide whether there was another reason for the overly pessimistic projections from the complex models used by MAFF. Or whether a performance like the earlier logistic fits to the data could have been achieved with more aggressive predation on the virus.

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