Immune-system models
Mathematical immune-system models describe interactions between a biological challenge and the host response. Depending on the question, the challenge may be a pathogen, infected cells or tumour cells, while the response may include innate cells, antibodies, T cells, cytokines or immune memory.
Begin with pathogen growth
Let \(P(t)\) be pathogen abundance. Without immunity, a simple early-growth model is
\[\boxed{\frac{dP}{dt}=rP,}\]where \(r\) is the pathogen net per-capita growth rate.
Its solution is
\[P(t)=P_0e^{rt}.\]This cannot describe immune control because nothing in the equation responds to the growing pathogen.
Add an immune effector population
Let \(E(t)\) denote an immune effector population. A simple interaction model is
\[\boxed{\frac{dP}{dt}=rP-kEP,}\]\[\boxed{\frac{dE}{dt}=s(P)-\delta E.}\]The term \(kEP\) represents immune-mediated pathogen removal, while \(s(P)\) represents pathogen-dependent immune stimulation.
Why the interaction term is a product
The term
\[kEP\]increases when there are more pathogens and also when there are more immune effectors. It is therefore a simple mass-action representation of encounters or effective interactions between the two populations.
It is an approximation, not a claim that every biological immune interaction literally follows mass action.
When does the pathogen increase or decrease?
Factor the pathogen equation:
\[\frac{dP}{dt}=P(r-kE).\]For \(P>0\), the sign is controlled by \(r-kE\). Therefore
\[\boxed{E<\frac{r}{k}\Rightarrow P\text{ increases},}\]\[\boxed{E>\frac{r}{k}\Rightarrow P\text{ decreases}.}\]The quantity \(r/k\) is an immune-effector threshold in this simple model.
A saturating immune-stimulation function
Immune activation does not necessarily increase indefinitely with pathogen abundance. A simple saturating function is
\[\boxed{s(P)=s_{max}\frac{P}{K_P+P}.}\]When \(P=K_P\), stimulation is half of its maximum:
\[s(K_P)=\frac{s_{max}}2.\]Pathogen–immune feedback through time
Using the saturating stimulation function gives
\[\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)-kEP,\]\[\frac{dE}{dt}=s_{max}\frac{P}{K_P+P}-\delta E.\]The logistic term prevents unlimited pathogen growth in this teaching example. The system is nonlinear because the variables interact.
Why the immune response can peak later
The pathogen is already present at the beginning, but the effector population needs time to be stimulated and accumulate. This creates a natural delay even without inserting an explicit time-delay term.
After pathogen abundance falls, stimulation weakens and effector loss \(\delta E\) can dominate, causing the immune response to contract.
Innate and adaptive immunity
A more detailed model can separate a rapidly acting innate response \(I(t)\) from a slower adaptive response \(A(t)\):
\[\frac{dP}{dt}=G(P)-k_IIP-k_AAP,\]\[\frac{dI}{dt}=F_I(P)-\delta_I I,\]\[\frac{dA}{dt}=F_A(P,A)-\delta_A A.\]The equations can encode different activation times, strengths and persistence.
Target cells and infected cells
For within-host viral infection, it is often useful to distinguish uninfected target cells \(T\), infected cells \(I\), and free virus \(V\):
\[\frac{dT}{dt}=\lambda-dT-\beta TV,\]\[\frac{dI}{dt}=\beta TV-\delta I,\]\[\frac{dV}{dt}=pI-cV.\]The infection term \(\beta TV\) transfers cells from the uninfected to infected class.
Interpreting the within-host terms
| Term | Interpretation |
|---|---|
| \(\lambda\) | production of target cells |
| \(dT\) | loss of target cells |
| \(\beta TV\) | new cellular infection |
| \(\delta I\) | loss of infected cells |
| \(pI\) | production of virus by infected cells |
| \(cV\) | clearance of free virus |
Add immune killing of infected cells
If \(E(t)\) represents cytotoxic effectors, the infected-cell equation can be extended to
\[\boxed{\frac{dI}{dt}=\beta TV-\delta I-kEI.}\]The term \(kEI\) represents additional loss of infected cells caused by the immune response.
Antibody models
Let \(B(t)\) represent an effective antibody concentration or abundance. A simple neutralisation term may be written
\[\frac{dV}{dt}=pI-cV-qBV.\]Here \(qBV\) represents antibody-associated removal or neutralisation of free virus in the model.
Cytokines
Cytokines act as signalling molecules between immune components. A simple cytokine variable \(C(t)\) might satisfy
\[\frac{dC}{dt}=p_CI-d_CC,\]where infected or activated cells stimulate cytokine production and \(d_C\) represents clearance.
Cytokine concentration can then feed back into immune-cell activation.
Immune expansion
Adaptive immune cells can proliferate after stimulation. One simple saturating expansion term is
\[\boxed{\frac{dE}{dt}=aE\frac{P}{K_P+P}-\delta E.}\]Unlike a source term that creates effectors independently of current \(E\), this form represents antigen-dependent proliferation of an existing effector population.
Immune contraction
When pathogen stimulation disappears, the expansion term becomes small and
\[\frac{dE}{dt}\approx-\delta E.\]The effector population then declines approximately exponentially.
Immune memory
After an acute response, some activated cells can enter a longer-lived memory compartment \(M(t)\):
\[\frac{dM}{dt}=\rho E-\mu M.\]A subsequent challenge can allow memory cells to contribute to a faster or stronger response.
Primary and secondary responses
A model with memory can produce different responses to first and later exposures. The second response may begin from a larger pool of responsive cells or have altered activation parameters.
This allows mathematical investigation of vaccination and immune recall at the within-host level.
Immune exhaustion
During persistent stimulation, immune effectiveness may decline. A model can include an exhaustion variable or divide effector cells into functional and exhausted states.
For example, a functional effector population might be lost to exhaustion at a pathogen-dependent rate:
\[\frac{dE}{dt}=\text{activation}-\delta E-\eta(P)E.\]Immune regulation
The immune system contains inhibitory as well as activating feedback. Regulatory cells or anti-inflammatory signals can suppress effector expansion.
Without negative regulation, a model may unrealistically predict uncontrolled immune activation.
Immunopathology
Reducing pathogen burden is not the only possible outcome of interest. A strong immune response can itself contribute to tissue damage.
A simple damage variable \(D(t)\) might satisfy
\[\frac{dD}{dt}=a_PP+a_EE-\gamma D.\]This separates pathogen-associated damage from immune-associated damage.
Thresholds and clearance
In deterministic continuous models, pathogen abundance can approach zero without literally reaching zero in finite time. Biological clearance, however, concerns discrete pathogen numbers.
When extinction is important, stochastic models are often more appropriate.
Stochastic immune models
At small population sizes, infection, cell activation, killing and extinction are discrete random events. Continuous-time Markov chains or stochastic reaction models can represent these events explicitly.
They can answer probability questions such as the chance that an infection establishes or is cleared.
Between-host and within-host models are different
Within-host models describe pathogen and immune dynamics inside an individual. Epidemic models describe transmission between individuals.
The variables, parameters and timescales are therefore different, even though both areas may use similar mathematical tools.
Linking within-host and epidemic scales
In multiscale models, within-host pathogen burden may influence infectiousness, which then affects transmission between hosts.
This connection must be defined explicitly rather than assuming that pathogen load and transmission rate are identical quantities.
Equilibria
An equilibrium satisfies
\[\frac{dP}{dt}=0,\qquad\frac{dE}{dt}=0.\]Possible equilibria may represent pathogen clearance, persistent infection or other long-term states depending on the model structure and parameters.
Stability
After finding equilibria, the Jacobian matrix can be used to study their local stability. Eigenvalues indicate whether small perturbations tend to decay or grow.
This connects immune-system modelling directly with dynamical-systems theory.
Parameter estimation
Parameters may be estimated from measurements such as pathogen load, immune-cell counts, antibodies or cytokines. Different quantities may be observed on different schedules and with different measurement errors.
A statistical observation model is therefore usually needed alongside the biological ODEs.
Identifiability
Complex immune models can contain many parameters that cannot all be determined reliably from limited data.
Adding biological detail does not automatically improve inference. A simpler identifiable model can be more useful than a detailed model whose parameters cannot be estimated.
A practical modelling workflow
Define the biological challenge and the immune component of interest. Start with the smallest interaction model that can answer the question. Write each biological process as a mathematical term and check its units. Analyse thresholds and equilibria. Solve the equations numerically when necessary. Compare trajectories with longitudinal data. Add innate/adaptive separation, antibodies, memory, regulation or stochasticity only when required by the question and supported by data.