Within-host infection models
Within-host infection models describe what happens after a pathogen enters one host. They follow processes such as infection of susceptible cells, pathogen replication, infected-cell loss, immune control and treatment.
Within-host is not the same as epidemic modelling
An epidemic model may use
\[S(t),\ I(t),\ R(t)\]for numbers of susceptible, infectious and recovered people.
A within-host model instead uses quantities such as susceptible cells, infected cells and pathogen abundance inside one individual.
The basic target-cell model
Let
\[T(t)=\text{uninfected target cells},\]\[I(t)=\text{infected cells},\]\[V(t)=\text{free pathogen or virus}.\]A standard model is
\[\boxed{\frac{dT}{dt}=-\beta TV,}\]\[\boxed{\frac{dI}{dt}=\beta TV-\delta I,}\]\[\boxed{\frac{dV}{dt}=pI-cV.}\]Read the model as biological flows
The first arrow represents infection of target cells. Infected cells then generate free pathogen, while free pathogen is cleared.
The infection term
The term
\[\boxed{\beta TV}\]depends on both available target cells and pathogen abundance. If either is zero, this mechanism produces no new infected cells.
The parameter \(\beta\) controls the strength of infection in this mass-action approximation.
Conservation between target and infected cells
Notice that \(\beta TV\) appears with opposite signs:
\[\frac{dT}{dt}=-\beta TV,\qquad \frac{dI}{dt}=+\beta TV-\delta I.\]An infection event removes a cell from the target class and adds it to the infected class. This bookkeeping is fundamental in compartment models.
Loss of infected cells
The term
\[\delta I\]represents loss of infected cells. If no new infection occurred,
\[\frac{dI}{dt}=-\delta I\]would give exponential decline with mean timescale \(1/\delta\).
Pathogen production and clearance
Infected cells produce free pathogen at rate \(pI\), while free pathogen is removed at rate \(cV\):
\[\frac{dV}{dt}=pI-cV.\]Thus free-pathogen abundance rises when production exceeds clearance and falls when clearance exceeds production.
What creates a pathogen peak?
Early in infection there may be many available target cells, so infection and pathogen production can dominate. Later, target-cell depletion, infected-cell loss, immunity or treatment can reduce production relative to clearance.
The peak occurs when
\[\boxed{\frac{dV}{dt}=0,}\]which in the basic model means
\[\boxed{pI=cV.}\]Model-generated infection trajectory
Why the variables peak at different times
The variables are connected sequentially. Target cells become infected, infected cells accumulate, and those infected cells produce pathogen. Each process has its own rate.
Therefore the maxima of \(I(t)\) and \(V(t)\) need not occur at exactly the same time.
A model with target-cell renewal
For longer infections, target cells may be produced and naturally lost:
\[\frac{dT}{dt}=\lambda-dT-\beta TV,\]\[\frac{dI}{dt}=\beta TV-\delta I,\]\[\frac{dV}{dt}=pI-cV.\]Without infection, the target-cell equilibrium is
\[\boxed{T_0=\frac{\lambda}{d}.}\]The within-host reproduction number
Near the infection-free state, one infected cell survives on average for time \(1/\delta\) and produces pathogen at rate \(p\). Each pathogen survives on average for time \(1/c\) and infects target cells at rate approximately \(\beta T_0\).
This gives the threshold quantity
\[\boxed{R_0^{within}=\frac{\beta pT_0}{c\delta}.}\]Interpretation of the threshold
In the standard target-cell model,
\[R_0^{within}>1\]means infection can initially grow near the infection-free equilibrium, while
\[R_0^{within}<1\]means the infection cannot maintain growth there.
How parameters affect the threshold
The expression
\[R_0^{within}=\frac{\beta pT_0}{c\delta}\]shows the model mechanisms directly. Stronger cellular infection \(\beta\), greater pathogen production \(p\), or more target cells \(T_0\) favour infection. Faster pathogen clearance \(c\) or infected-cell loss \(\delta\) oppose it.
Threshold as a function of infection rate
Add an immune response
Let \(E(t)\) represent an immune effector population. Immune killing of infected cells can be represented by
\[\frac{dI}{dt}=\beta TV-\delta I-kEI.\]An effector equation might be
\[\frac{dE}{dt}=s(I)-\mu E,\]where \(s(I)\) represents stimulation by infection.
Direct pathogen neutralisation
An immune component such as an effective antibody variable \(A(t)\) may instead act on free pathogen:
\[\frac{dV}{dt}=pI-cV-qAV.\]Thus immune mechanisms can enter different parts of the infection cycle.
Innate and adaptive responses
Innate responses often act earlier, while adaptive responses may expand more slowly and persist differently. A model can represent them with separate variables and timescales.
This can produce delayed control without imposing an arbitrary shape on the response.
Latent or eclipse stages
A newly infected cell may not produce pathogen immediately. Introduce an eclipse compartment \(E\):
\[T\xrightarrow{\beta TV}E\xrightarrow{k}I.\]Then
\[\frac{dE}{dt}=\beta TV-kE,\]\[\frac{dI}{dt}=kE-\delta I.\]The additional state creates a biologically interpretable delay between infection and pathogen production.
Drug treatment
Antiviral treatment can act on different processes. For example, a treatment reducing new cellular infection by efficacy \(\varepsilon_\beta(t)\) gives
\[\beta TV\longrightarrow[1-\varepsilon_\beta(t)]\beta TV.\]A treatment reducing pathogen production can give
\[pI\longrightarrow[1-\varepsilon_p(t)]pI.\]Link treatment to pharmacokinetics
Instead of assuming constant efficacy, drug concentration can be modelled explicitly:
For example,
\[\varepsilon(C)=\frac{E_{max}C}{EC_{50}+C}.\]This couples pharmacokinetics, pharmacodynamics and within-host infection dynamics.
Why treatment timing matters
The state of the system changes through infection. Early treatment may act while many target cells remain and pathogen is expanding; later treatment acts on a different combination of target cells, infected cells and pathogen.
Timing effects should therefore emerge from the dynamical equations rather than being imposed as a hand-drawn trajectory.
Drug resistance
Separate sensitive and resistant pathogen populations can be introduced:
\[V_S(t),\qquad V_R(t).\]They may differ in replication, infection or treatment-response parameters. Mutation can transfer probability or abundance between types.
Treatment can then alter not only total pathogen burden but also population composition.
Chronic infection
With target-cell renewal, immune regulation or persistent reservoirs, the model can admit a positive infected equilibrium rather than complete clearance.
Equilibrium analysis helps distinguish parameter regimes associated with clearance and persistence.
Equilibria
For the renewal model, an infection-free equilibrium is
\[\boxed{(T,I,V)=\left(\frac{\lambda}{d},0,0\right).}\]Positive equilibria are found by setting all derivatives equal to zero and solving the resulting algebraic equations.
Stability
The Jacobian matrix describes how small perturbations evolve near an equilibrium. Eigenvalues can determine whether the infection-free state is locally stable or unstable.
The threshold \(R_0^{within}=1\) is closely connected with this change in stability in the standard model.
Stochastic within-host models
When pathogen or infected-cell numbers are small, deterministic ODEs can miss extinction by chance.
A continuous-time Markov chain can treat infection, production, clearance and cell death as discrete random events. It can then estimate probabilities of establishment or clearance.
Why stochasticity matters near the start
Even when a deterministic model predicts initial growth, a very small inoculum may disappear by chance before establishing a large infection.
Thus a deterministic growth threshold and a probability of establishment answer related but different questions.
Spatial within-host models
Infection is not always well mixed. Pathogen and immune cells can vary across tissues.
A spatial model may use pathogen density \(V(x,t)\) and include diffusion or transport:
\[\frac{\partial V}{\partial t}=D\nabla^2V+\text{production}-\text{clearance}.\]This can represent local spread and heterogeneous tissue environments.
Connecting within-host dynamics to infectiousness
A multiscale model may define infectiousness as a function of within-host pathogen burden:
\[\boxed{\lambda_{trans}(t)=f(V(t)).}\]The function \(f\) must be specified and estimated; viral load and transmission rate should not simply be assumed identical.
From one host to an epidemic model
This creates a bridge between cellular-scale infection models and population epidemiology.
Observation models
Experiments usually measure only some state variables, often with error. A measured pathogen load \(Y(t)\) might be represented as
\[Y(t)=h(V(t))+\text{measurement error}.\]The observation process should be distinguished from the underlying biological state equation.
Parameter estimation
Parameters such as infection rate, infected-cell loss and pathogen clearance can be estimated from longitudinal measurements, but not every parameter is identifiable from every dataset.
Measurements of several biological quantities can provide more information than pathogen load alone.
Identifiability
Different parameter combinations may produce similar pathogen trajectories. A visually good fit therefore does not guarantee that each biological rate has been estimated reliably.
Structural and practical identifiability should be considered before interpreting parameters mechanistically.
A practical modelling workflow
Define the biological scale and state variables. Draw the biological transitions conceptually, then translate each transition into a mathematical rate. Check signs and units. Identify thresholds and equilibria. Solve the ODEs numerically and compare model-generated trajectories with data. Add immunity, treatment, latency, resistance, spatial structure or stochasticity only when required by the biological question.