Host–parasite systems
A parasite obtains resources from a host and usually reduces host fitness. At population level, the key processes are transmission, recovery or parasite loss, host demography and parasite-induced harm.
Susceptible and infected hosts
Let
\[S(t)=\text{susceptible hosts},\qquad I(t)=\text{infected hosts},\qquad N(t)=S(t)+I(t).\]Under density-dependent transmission, new infections occur at rate
\[\beta SI.\]The same event removes one susceptible host and creates one infected host, so the transmission term enters the two equations with opposite signs.
A density-regulated host–parasite model
To keep the host population biologically bounded, suppose total host recruitment is logistic:
\[rN\left(1-\frac{N}{K}\right).\]Let infected hosts recover at rate \(\gamma\), and let parasitism cause additional mortality at rate \(\alpha\). Then
\[\boxed{\frac{dS}{dt}=rN\left(1-\frac{N}{K}\right)-\beta SI+\gamma I},\]\[\boxed{\frac{dI}{dt}=\beta SI-(\gamma+\alpha)I}.\]| Symbol | Meaning |
|---|---|
| \(r\) | host population growth parameter |
| \(K\) | host carrying capacity in the absence of parasite-induced mortality |
| \(\beta\) | transmission coefficient |
| \(\gamma\) | recovery or parasite-clearance rate |
| \(\alpha\) | additional host mortality caused by parasitism |
What the total host equation says
Adding the two equations gives
\[\boxed{\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)-\alpha I}.\]Transmission and recovery cancel because they only move hosts between infection states. Parasite-induced mortality reduces the total host population.
When can infection increase?
The infected-host equation factors as
\[\frac{dI}{dt}=I\left[\beta S-(\gamma+\alpha)\right].\]Therefore a rare infection initially increases when
\[\beta S>\gamma+\alpha.\]The critical susceptible abundance is
\[\boxed{S_c=\frac{\gamma+\alpha}{\beta}}.\]Basic reproduction number
At the parasite-free equilibrium of this density-regulated host model, \(S_0=K\). The invasion quantity is therefore
\[\boxed{\mathcal R_0=\frac{\beta K}{\gamma+\alpha}}.\]If \(\mathcal R_0>1\), a rare infection can increase deterministically. If \(\mathcal R_0<1\), it declines when introduced near the parasite-free equilibrium.
Why parasite burden can matter
A susceptible/infected model treats infection as a binary state. This is often appropriate for microparasites, but many macroparasites are better described by the number of parasites carried by each host.
Parasite aggregation can then matter: the same mean burden can correspond to nearly equal burdens across hosts or to a few heavily infected hosts and many uninfected hosts.
Density-dependent and frequency-dependent transmission
The term \(\beta SI\) is density-dependent. If contact per host does not rise strongly with host density, a frequency-dependent form such as
\[\beta\frac{SI}{N}\]may be more appropriate. This changes both the units of \(\beta\) and the invasion threshold.
Environmental stages and complex life cycles
Some parasites use environmental stages, vectors or intermediate hosts. If \(E(t)\) denotes infectious material in the environment, a model may include
\[\frac{dE}{dt}=\text{release from hosts}-\text{environmental loss},\]with host infection depending on exposure to \(E\) rather than directly on \(I\).
Virulence and transmission
In simple mortality-based models, \(\alpha\) measures parasite-induced host mortality. Increasing virulence can shorten the infectious period, but in some biological systems more intense exploitation can also increase transmission. Evolutionary models study this trade-off explicitly.
Connection with epidemiology
Many epidemic models are host–parasite models in a broad sense. The same ideas of transmission, recovery, mortality, invasion thresholds and \(\mathcal R_0\) appear there, while ecological models often place more emphasis on host demography, parasite burden and life-cycle structure.
Stochastic extinction
When infected hosts are rare, transmission and recovery are discrete random events. Even when \(\mathcal R_0>1\), chance can eliminate the infection before a large outbreak or persistent parasite population develops.
A continuous-time Markov chain can represent these events through transition rates and calculate or simulate extinction probabilities.