Density-Dependent and Frequency-Dependent Transmission

The rate at which new infections appear is a product of how many infectious individuals there are and how often a susceptible meets one. That second piece — the contact rate — is where two classic assumptions part ways: does an individual meet more others as the population gets denser, or does it keep a roughly fixed number of contacts no matter how crowded things get? The answer decides whether a pathogen faces a critical host density below which it cannot spread, and it changes what control does.

One transmission term, two contact rules#

Write new infections per unit time as a per-capita contact rate $C(N)$, a per-contact transmission probability $\beta$, and the chance a contact is with an infectious individual, $I/N$: $$\text{new infections} = \beta\,C(N)\,S\,\frac{I}{N}.$$ Everything hinges on how the contact rate $C(N)$ depends on the total host density $N$.

Density-dependent transmission assumes contacts scale with density, $C(N)=\kappa N$. Twice as many hosts in the same area means twice as many encounters per individual. Substituting collapses the term to the mass-action form $$\frac{dI}{dt}=\beta\kappa\,S I - \gamma I,$$ where the transmission constant is often just written $\beta' = \beta\kappa$. This is the assumption baked into the classic SIR model when it is written with $\beta S I$.

Frequency-dependent transmission (also called standard incidence) assumes the contact rate is fixed, $C(N)=\kappa$, independent of density. Each individual makes about the same number of contacts whether the population is sparse or dense. The term becomes $$\frac{dI}{dt}=\beta\kappa\,\frac{S I}{N} - \gamma I,$$ which is the $\beta S I / N$ form.

The two models look almost identical — they differ only by a factor of $N$ — but that factor changes the biology completely.

The critical difference: a density threshold#

Consider a pathogen invading a fully susceptible population, so $S\approx N$. It spreads when $dI/dt>0$, i.e. when the basic reproduction number $R_0>1$.

Under density dependence, seeding one infection into $S=N$ gives $dI/dt = (\beta\kappa N - \gamma)I$, so $$R_0 = \frac{\beta\kappa N}{\gamma}.$$ This scales with density. Setting $R_0=1$ gives a critical host density $$N_T = \frac{\gamma}{\beta\kappa},$$ below which the pathogen cannot invade no matter how it is introduced. Thinning the host population below $N_T$ eradicates the disease — the logic behind culling and behind the local fadeout of measles in small communities before vaccination.

Under frequency dependence, the $N$ cancels: $$R_0 = \frac{\beta\kappa}{\gamma},$$ independent of population size. There is no density threshold. A sexually transmitted infection can persist in a small, sparse population just as well as a large one, because partners are acquired at a rate set by behavior, not by crowding.

Show how β and R₀ change with population size

Both models are the same equation with a different transmission coefficient. Write new infections in the standard-incidence form $\beta(N)\,S\,I/N$ and let a single coefficient $\beta(N)=\beta\,C(N)$ absorb the contact rule. The two assumptions are then just two shapes for that coefficient:

$$\beta(N) = \beta\kappa N \quad\text{(density-dependent)}, \qquad \beta(N) = \beta\kappa \quad\text{(frequency-dependent)}.$$

Under density dependence the coefficient grows linearly with crowding; under frequency dependence it is flat. The power-law compromise $C(N)=\kappa N^{q}$ interpolates the two as $\beta(N)=\beta\kappa N^{q}$.

Seed one infection into a fully susceptible population, $S\approx N$. The infected class then obeys

$$\frac{dI}{dt} = \left[\beta(N)\,\frac{S}{N} - \gamma\right]I = \big[\beta(N) - \gamma\big]I,$$

so the basic reproduction number is $R_0 = \beta(N)/\gamma$, giving

$$R_0^{\mathrm{DD}} = \frac{\beta\kappa N}{\gamma}, \qquad R_0^{\mathrm{FD}} = \frac{\beta\kappa}{\gamma}.$$

The density-dependent $R_0$ carries a factor of $N$; the frequency-dependent one does not. The invasion (per-capita) growth rate is $r(N)=\beta(N)-\gamma=\gamma\,(R_0-1)$, so it shares the sign of $R_0-1$:

$$r_{\mathrm{DD}}(N) = \beta\kappa N - \gamma, \qquad r_{\mathrm{FD}}(N) = \beta\kappa - \gamma.$$

The density-dependent rate is a rising line that crosses zero exactly at the critical density $N_T=\gamma/(\beta\kappa)$ — below it the pathogen shrinks, above it it grows — while the frequency-dependent rate is a horizontal line whose sign never depends on $N$.

Which assumption fits which disease#

The right choice is an empirical question about how contact scales, not a mathematical preference.

• Density-dependent fits many directly transmitted diseases where crowding raises encounter rates: airborne and fecal-oral infections in wildlife and livestock, phocine distemper in seals, many plant pathogens.
• Frequency-dependent fits sexually transmitted infections (partner acquisition is behavioral), vector-borne diseases (the biting rate is set by the vector, not host density), and social-contact diseases in populations that regulate group size.

Real systems often sit between the two, and a common compromise is the power law $C(N)=\kappa N^{q}$ with $0\le q\le 1$: $q=1$ recovers density dependence, $q=0$ recovers frequency dependence. Fitting $q$ to contact or incidence data is a standard way to let the data decide (McCallum, Barlow & Hone 2001).

A worked example#

Take a recovery rate $\gamma=0.1\ \text{day}^{-1}$ (a 10-day infectious period) and compare two population sizes, $N=100$ and $N=1000$.

For a density-dependent pathogen with $\beta\kappa=0.002\ \text{day}^{-1}$ per individual, the threshold is $N_T=\gamma/(\beta\kappa)=0.1/0.002=50$ hosts. So $R_0=\beta\kappa N/\gamma$ is $2$ at $N=100$ and $20$ at $N=1000$ — the same pathogen barely spreads in the small population and explodes in the large one.

For a frequency-dependent pathogen with $\beta\kappa=0.3\ \text{day}^{-1}$, $R_0=0.3/0.1=3$ in both populations. Halving or doubling the host density does nothing to invasion.

Watching the threshold in an SIR epidemic#

The invasion algebra becomes vivid when we run the full SIR model forward. Keep $\gamma=0.1\ \text{day}^{-1}$ and the two transmission constants from above, seed one infection, and compare population sizes $N=100$ and $N=500$.

A density-dependent epidemic uses the mass-action term $\beta\kappa\,S I$:

$$\frac{dS}{dt} = -\beta\kappa\,S I, \qquad \frac{dI}{dt} = \beta\kappa\,S I - \gamma I, \qquad \frac{dR}{dt} = \gamma I,$$

so $R_0=\beta\kappa N/\gamma$ climbs from $2$ at $N=100$ to $10$ at $N=500$.

A frequency-dependent epidemic divides the same term by $N$:

$$\frac{dS}{dt} = -\beta\kappa\,\frac{S I}{N}, \qquad \frac{dI}{dt} = \beta\kappa\,\frac{S I}{N} - \gamma I, \qquad \frac{dR}{dt} = \gamma I,$$

so $R_0=\beta\kappa/\gamma=3$ at both sizes.

Colour marks the contact rule and dashing marks the population size, so each curve is one row of the earlier $R_0$ table playing out in time. The two frequency-dependent curves (orange) are the same epidemic: both have $R_0=3$, so they share a growth rate and reach the same fraction of the population — raising $N$ from 100 to 500 just scales the count up about fivefold (a single seed is a smaller starting fraction in the larger population, so its wave arrives a little later). The two density-dependent curves (blue) are different epidemics: at $N=100$ the pathogen barely spreads ($R_0=2$, a slow bump peaking near 16), while at $N=500$ it explodes ($R_0=10$, a fast wave peaking above 330). Raising the density changes what kind of outbreak happens under density dependence, but only how large it is under frequency dependence.

In code#

We compute $R_0$ under both assumptions at two densities to see the density dependence appear and disappear.

R#

gamma <- 0.1

# density-dependent: R0 scales with N, threshold at N_T
bk_dd <- 0.002
R0_dd <- function(N) bk_dd * N / gamma
N_T <- gamma / bk_dd                 # critical host density = 50

# frequency-dependent: R0 independent of N
bk_fd <- 0.3
R0_fd <- function(N) bk_fd / gamma

sapply(c(100, 1000), R0_dd)          # 2 and 20
sapply(c(100, 1000), R0_fd)          # 3 and 3
N_T                                  # 50

Python#

gamma = 0.1
bk_dd, bk_fd = 0.002, 0.3

R0_dd = lambda N: bk_dd * N / gamma          # scales with density
R0_fd = lambda N: bk_fd / gamma              # flat in density
N_T = gamma / bk_dd                          # critical host density

print("density-dependent R0:", [round(R0_dd(N), 2) for N in (100, 1000)])
print("frequency-dependent R0:", [round(R0_fd(N), 2) for N in (100, 1000)])
print("critical host density N_T:", N_T)

density-dependent R0: [2.0, 20.0]
frequency-dependent R0: [3.0, 3.0]
critical host density N_T: 50.0

Julia#

gamma = 0.1
bk_dd, bk_fd = 0.002, 0.3

R0_dd(N) = bk_dd * N / gamma      # scales with density
R0_fd(N) = bk_fd / gamma          # flat in density
N_T = gamma / bk_dd               # critical host density = 50

R0_dd.((100, 1000))               # (2.0, 20.0)
R0_fd.((100, 1000))               # (3.0, 3.0)

Why it matters#

The transmission term is the single most consequential modeling choice in an epidemic model, and it is easy to make by accident: writing $\beta S I$ versus $\beta S I / N$ commits you to a whole theory of how contact works. Get it wrong and the model can predict a density threshold that does not exist, or miss one that does — which in turn misprices interventions like culling, vaccination, and social distancing. The same distinction propagates into $R_0$ calculations via the next-generation matrix, into the vector-borne models where frequency dependence is the norm, and into the evolution of virulence, where how transmission scales with host density shapes the selective pressure on the pathogen.

Related#

• Compartmental Models (SIR) — where the $\beta S I / N$ term comes from
• The Next-Generation Matrix and R₀
• Vector-Borne Disease Models — frequency-dependent transmission via biting
• SEIR and Compartmental Extensions
• Exponential and Logistic Growth — density dependence in population growth
• Adaptive Dynamics and the Evolution of Virulence
• Quantitative Methods