Density-dependent vs. Frequency-dependent Disease Transmission

(The difference between density-dependent (DD) and frequency-dependent (FD) disease transmission is universally difficult to understand, especially at a fundamental level.  It doesn’t help that there has been debate in the scientific literature about the use of these terms, and the underlying arguments for that debate are less-than-transparent for people who aren’t particularly interested/fluent in the math.  In this post, I’m going to attempt to explain the underlying assumptions of the two models while following Begon et al. (2002), who have quite possibly written one of the world’s best examples of a math-centric paper that non-math people can understand.  You can access a PDF of their wonderful paper here.)

The Fundamental Difference Between DD and FD Transmission:

For those of you who are only here in attempt to quickly google the difference between DD and FD transmission, I’m going to give you the answer up front, and then derive/explain below.

In density-dependent transmission, the per capita contact rate between susceptible (S) and infected (I) individuals depends on the population density.  So, transmission rates increase with density.

In frequency-dependent transmission, the per capita contact rate between susceptible (S) and infected (I) individuals does not depend on the population density.  So transmission rates do not change with density.

Beta:  The Transmission Rate

When using differential equations to model pathogen/parasite transmission, the most important part of the model is arguably the ‘transmission term.’  Ignoring all other model terms, the classic equations for DD and FD transmission are:

DD:  dI/dt = B * S * I

FD:  dI/dt = B’ * S * I/N

Where S is the number of susceptible individuals, I is the number of infected individuals, and N is the total number of individuals in the population.  (If you’re new to differential equations, the dI/dt part says “the rate of change in the number of infected individuals with respect to time is equal to….”)  You’ll notice that for DD, there is a variable B, and for FD, there is a variable B’.  Both of these variables are transmission rates.  However, I use the prime to indicate that while B and B’ can be equivalent under some circumstances, B =/= B’, which is a very important distinction that most textbooks do not make.

We now need to answer two questions about how and why the equations for DD and FD transmission are different.  It will take some time to build an explanation, but don’t forget that answering these questions is the goal of this post:

  1. How and why are B and B’ different?
  2. Why does that DD equation have just an I, while the FD equation has an I/N?

Force of Infection:

This is a good time to introduce the force of infection, which is something that I have previously blogged about without explaining it very well.  When we look at those dI/dt equations, we’re asking, “What is the rate at which we increase the number of infected individuals in the population?”  Or, equivalently, “What is the per capita rate at which susceptible individuals get infected?”  That rate has a special name – it is the force of infection (FOI).  FOI is a rate that is pretty easy to measure in some systems; you just stick susceptible animals out in the field and count how many get infected per unit time.  Another way to look at those transmission equations is:

DD:  dI/dt = λ * S

FD:  dI/dt = λ’ * S

Where λ and λ’ are the force of infection terms.  Again, the prime means that λ =/= λ’.

I like to call the force of infection a ‘conglomerate’ rate because it is really a combination of other rates.  That is, FOI is the product of three probabilities/rates: 1) the probability/rate that a contact happens between individuals in the population (c or c’), 2) the probability/rate that a given contact is with an infected individual (I/N), and 3) the probability/rate that a contact between an S and an I individual results in successful transmission of the parasite (v).  (I’m using Begon et al.’s (2002) variables for these rates.)  So, we can say that:

DD:  FOI = λ = c * v * I/N

FD:  FOI = λ’ = c’ * v * I/N

As you can see, the reason that λ =/= λ’ is because c =/= c’.  Furthermore, remember the transmission rates B and B’?  Well, B = c*v and B’ = c’ * v.  So the reason that B =/= B’ is also because c =/= c’.  We’ve partially answered Question 1!  But before we can go any further with our transmission equations, we need to figure out why the c and c’ are different.  Remember that these rates are the probabilities/rates that a single contact happens between individuals in the population – for short, we call them contact rates.

The DD and FD Contact Rates are Different:

In DD transmission, the contact rate (c) depends on the population density.  In FD transmission, the contact rate (c’) does not depend on the population density.  This is the main difference between density-dependent and frequency-dependent transmission.  Sometimes it is easier to say those two sentences than to truly understand what they mean, so let’s do a little thought experiment:

Dr. Green is going clubbing.  Dr. Green can either go to Club 1 or Club 2.  It is flu season, and while deciding which club to frequent, Dr. Green wonders whether Club 1 or Club 2 is better for avoiding disease transmission.  (I’m sure everyone uses this as the criteria for finding a good club.)  The two clubs are exactly the same size (A1 = A2 = 1), but Club 2 always has more people than Club 1 (N1 < N2; i.e., Club 2 has a higher density).  Dr. Green realizes that the contact rate with other people will be higher in Club 2 because the density is greater.  That is, the contact rate (c) is density-dependent.  The more people in the club, the more likely Dr. Green is to get close enough to one of them to acquire some of their germs.

DrGreenGoesClubbing

In the previous scenario, we were talking about the types of contacts that lead to transmission of the flu, rhinovirus, etc.  What if we were talking about STDs instead?  Does the probability of contact leading to an STD infection (i.e., the probability of having sex with one of the club-goers) increase with the density of people in the club?  No (for the most part).  If Dr. Green is going to have sex with one of the club-goers, it is unlikely to depend on whether there are 50 or 100 club-goers.  The contact rate (c’) is not density-dependent.  The caveat is that contact rate might be density-dependent at very low densities.  Like, if Dr. Green walks into the club and there are only two club-goers, both of whom have not bathed in a year, the probability of sexual contact will likely be much lower than if Dr. Green walks into a club with 50 club-goers.  (But hey, whatever floats your boat, Dr. Green.  We don’t judge here on Parasite Ecology.)

We can summarize what we’ve learned about contact rates in DD and FD transmission in a graph.  In DD transmission, the contact rate (c) increases with the population density, where density is just the total number of individuals in the population (N) divided by the area (A).  We usually assume that the relationship between c and N/A is linear, with a slope of k.  However, we don’t need to assume a linear relationship if a different shape makes more sense.  In FD transmission, the contact rate (c’) is usually not affected by the population density (N/A) – see the green dashed line.  That is, the line for c’ has an intercept of n and a slope of 0.  The caveat being that at in some cases, c’ does increase with density at very low densities – see the orange dashed line.

DD vs FD

We’ve officially answered Question 1!  That is, B and B’ are different because c and c’ are different.  Specifically, c is a function of density, and c’ is a constant.

Why does FD Transmission Have an I/N?

Finally, let’s answer Question 2:  why does the DD equation have an I, but the FD equation has an I/N?  Let’s go back to our equations, but instead of using the conglomerate FOI term, we’ll look at the rates c or c’, v, and I/N.

DD:  dI/dt =  c * v * I/N * S

FD:  dI/dt =  c’ * v * I/N * S

But remember that we just found line equations for c and c’, where c = k*N/A and c’ = n.  So, let’s plug in those line equations:

DD:  dI/dt =  k * N/A * v * I/N * S

FD:  dI/dt =  n * v * I/N * S

Things are getting a little messy, but we can simplify the DD equation because N occurs in both the numerator and the denominator.  (This is the key step to understanding Question 2!)  Simplifying gives us:

DD:  dI/dt =  k * 1/A * v * I * S

FD:  dI/dt =  n * v * I/N * S

We’re getting really close to the equations that we originally looked at in the beginning of this post.  But now, let’s lump some terms together and think about the transmission rates, B and B’.  In DD transmission, B = k*v.  In FD transmission, B’ = n*v.  Substituting in B and B’ and moving some stuff around, we now have:

DD:  dI/dt =  B * S * I/A

FD:  dI/dt =  B’ * S * I/N

TADA!  …no?  You aren’t satisfied?  You’re thinking, “Hey!  That DD equation has an I/A, instead of an I!  That isn’t right!”  Well, actually, it is correct, but people very commonly and wrongly don’t include the A.  However, if the area that a population occupies is constant and/or we’re comparing different populations who occupy the same sized areas, then A = 1 and we can leave the A out of the equation.  This brings us to the DD and FD equations that you all know and love:

DD:  dI/dt =  B * S * I

FD:  dI/dt =  B’ * S * I/N

Therefore, DD has an I while FD has an I/N because there is an N in the contact rate (c) for DD transmission that cancels with the N in the denominator.  Because c’ is not density-dependent, it does not have an N component, so the N in the denominator does not cancel out.

Some Notes about N and A:

This section is for more advanced readers.  You don’t really need to understand this to understand the basics of DD and FD transmission.

If you’re trying to decide whether DD or FD transmission make more sense for a given parasite/pathogen system, you need to decide whether contact rate is density-dependent or not.  If you’re modeling professionally, I recommend that you 1) read the Begon et al. (2002) paper as well as others (e.g., McCallum et al. 2001) and 2) consider N and A very carefully.

We very often assume that the area that a population occupies is constant, or that the areas occupied by multiple populations that we are comparing are equivalent.  Sometimes those assumptions are reasonable, and for DD transmission, we can say that dI/dt = B * S * I.  But those assumptions are often 1) not reasonable and 2) not even evaluated, so it is safer to start by thinking that dI/dt = B * S * I/A.

Here are two additional points of interest about N, A, and constant density:

If A is constant AND N is constant, then the density is constant.  And if the density is constant, both c and c’ are constants.  Furthermore, under this scenario, density-dependent and frequency-dependent transmission equations yield the same results.

If A is proportional to N – for instance, if the population occupies a larger area as it grows – then the density is constant, and FD transmission is the correct model.

Again, I highly recommend taking a look at Begon et al. (2002).  Let me know in the comments if anything was particularly unclear, and/or if you have suggestions for improving this post.

Also, if you’re wondering if we get more than two options – just FD and DD – check this out.

Force of Infection

I haven’t posted about a paper recently, so here’s a random grab out of my massive folder of “PDFs to Read Someday.”  (I should use differential equations to model how the number of PDFs in that folder changes over time…except that it appears to be exponentially increasing, so maybe I don’t want to know.) 

First, let’s talk about force of infection (FOI).  FOI is the rate at which susceptible individuals become infected per unit time.  It is an important parameter (the most important?) for transmission in epidemiological models.  The problem is that FOI isn’t as easy to measure as you might think.  In order to get good estimates of the rate of infection, you typically need to monitor infection status in many, many individuals.  For instance, even assuming you can identify infected individuals with 100% accuracy, your estimate of FOI is going to be better if you sample 100 individuals (1% resolution) than 20 individuals (5%) resolution. 

Movement rates between classes of the SIR model

Serious lack of FOI-related pictures on the Internet. Diagram from here.

But of course, you can’t identify infected individuals with 100% accuracy.  This is especially apparent when organisms have a period of latent infection, or when infection gets more obvious the longer the individual is infected.  For instance, if a snail has had first intermediate trematode infection for a while, crushing the snail might result in a mini-explosion of rediae/sporocysts all over your dissection surface.  (A brief aside for the curious.)  If it’s a newly developed infection, you might be lucky to spot a single redia/sporocyst.  The same is true when looking for Plasmodium falciparum (malaria parasite) in blood smears; you’re much more likely to identify an infection if parasitemia is very high.

What once were snail gonads are now trematode duplexes. Trematode sprawl. Photo from here.

Finally, if an individual can get infected multiple times, you would underestimate FOI if you could only identify the first infection event.  This is where the Mueller et al. (2012) paper from PNAS comes in.  For P. falciparum in Papua New Guinea, most people have infections with multiple different clones of P. falciparum.  That means that they got multiple clonal infections from one mosquito bite and/or they got subsequent infections with different clones from different mosquito bites. 

To get a super duper accurate FOI measurement with molecular techniques (molFOI), Mueller et al. (2012) took blood samples from ~400 kids aged 0-4 every two months for 69 weeks.  Impressive!  Then they used genotyping of the mps2 gene to determine how many P. falciparum clones each child had at each time point.

Cutting to the chase:  molFOI and the incidence of clinical cases of malaria (i.e., number of reported cases with high parasite loads) had very similar patterns across child age and week/season.  molFOI had a strong relationship with week/season, and this relationship explained ~50% of the relationship between malaria incidence and season.  Furthermore, most of the age*incidence relationship was explained by the age*molFOI relationship.  So, cool new tool for monitoring transmission in malaria control programs!  Neat!

Though they certainly aren’t the first people to demonstrate the efficacy of using insecticide treated bed nets, they also found that ITNs reduced infection risk by ~50%.  Awesome.

Multiple clonal infections are the norm in the P. falciparum system mentioned here.  Do you think that is true in many disease systems? 

Reference:

Muellera, I., S. Schoepflind, T.A. Smithd, et al. 2012. Force of infection is key to understanding the epidemiology of Plasmodium falciparum malaria in Papua New Guinean children.  PNAS 109:25, 10030-10035.