Multi-strain models
A multi-strain model describes a pathogen when two or more distinguishable strains or variants circulate in the same population. The strains may infect the same type of host but behave differently enough that treating them as one infection would hide important epidemic behaviour.
Start with an ordinary one-strain model
In a simple SIR model, all infectious individuals are placed in one compartment \(I\):
This is reasonable when the circulating pathogen can be treated as epidemiologically uniform.
But suppose two strains circulate and strain 2 transmits more easily than strain 1. If both are placed inside the same \(I\) compartment, the model cannot tell which strain is growing or replacing the other.
Step 1: separate infections by strain
For two strains, introduce two infectious compartments:
\[I_1(t)=\text{hosts infectious with strain 1},\] \[I_2(t)=\text{hosts infectious with strain 2}.\]A susceptible person can now become infected through either route.
The diagram is deliberately simple. It shows the first essential change: infection is no longer a single pathway.
Step 2: give each strain its own transmission parameters
Let \(\beta_1\) and \(\beta_2\) be the transmission-rate parameters for strains 1 and 2. Their forces of infection can be written
\[\lambda_1=\beta_1\frac{I_1}{N},\qquad \lambda_2=\beta_2\frac{I_2}{N}.\]A susceptible individual is exposed to both strains, so the susceptible equation becomes
\[\boxed{\frac{dS}{dt}=-\lambda_1S-\lambda_2S}.\]The two terms have clear meanings:
| Flow | Meaning |
|---|---|
| \(\lambda_1S\) | new infections caused by strain 1 per unit time |
| \(\lambda_2S\) | new infections caused by strain 2 per unit time |
A simple numerical interpretation
Suppose a population contains 10,000 susceptible people. At a particular time, let
\[\lambda_1=0.002\text{ day}^{-1},\qquad \lambda_2=0.006\text{ day}^{-1}.\]Then the instantaneous infection flows are
\[0.002(10000)=20\quad\text{strain-1 infections per day},\] \[0.006(10000)=60\quad\text{strain-2 infections per day}.\]Step 3: strains can differ in more than transmissibility
| Difference | Possible model representation |
|---|---|
| transmissibility | different \(\beta_1,\beta_2\) |
| infectious duration | different recovery rates \(\gamma_1,\gamma_2\) |
| latent period | different progression rates in an SEIR model |
| immune escape | different susceptibility after previous infection or vaccination |
| disease severity | different hospitalisation or mortality probabilities |
| response to intervention | strain-specific vaccine protection or treatment effect |
Therefore “strain 2 is different” should not automatically be represented only by changing \(\beta\). The parameter changed should correspond to the biological difference being modelled.
Step 4: understand competition for susceptible hosts
If both strains infect the same susceptible population, they are competing for a shared resource: susceptible hosts.
When strain 1 infects a susceptible person, that person is no longer immediately available to strain 2 under a model with complete post-infection protection. The reverse is also true.
This indirect interaction is called competition. The strains do not need to physically interact with each other; they influence one another by changing the host population available for infection.
Why a more transmissible strain can replace another
Suppose strain 2 has a larger effective reproduction number than strain 1 under the same population conditions. Its infections may increase more rapidly. As strain 2 occupies a larger fraction of new infections, the relative frequency of strain 1 can fall.
This figure shows replacement, not a universal rule. Whether replacement occurs depends on immunity, interventions, timing, introductions and other strain characteristics.
Step 5: previous infection may change susceptibility to another strain
The simplest model may assume that recovery from either strain gives complete protection against both. Real pathogens may not behave this way.
After recovering from strain 1, a person might have:
| Cross-immunity | Meaning |
|---|---|
| complete | strain 1 infection fully protects against strain 2 |
| partial | strain 1 infection reduces, but does not eliminate, susceptibility to strain 2 |
| none | strain 1 infection gives no protection against strain 2 |
This protection between different strains is called cross-immunity.
Step 6: represent partial cross-immunity
Let \(R_1\) denote people recovered from strain 1. Suppose \(\sigma_{12}\) represents their susceptibility to strain 2 relative to a fully susceptible person.
| Value | Interpretation |
|---|---|
| \(\sigma_{12}=0\) | complete protection against strain 2 |
| \(0<\sigma_{12}<1\) | partial protection |
| \(\sigma_{12}=1\) | no protection |
The strain-2 infection flow from this recovered group can then be represented as
\[\boxed{\sigma_{12}\lambda_2R_1}.\]Cross-immunity is directional
Protection need not be the same in both directions. We may need both
\[\sigma_{12}\quad\text{and}\quad\sigma_{21}.\]Here \(\sigma_{12}\) describes susceptibility to strain 2 after strain 1, whereas \(\sigma_{21}\) describes susceptibility to strain 1 after strain 2.
Sequential infection requires extra states
If people can recover from one strain and later acquire another, a single recovered compartment \(R\) is no longer enough. The model needs to remember infection history.
Here \(R_1\) and \(R_2\) retain information about which strain was experienced first, and \(R_{12}\) can represent people who have experienced both. More realistic models may require still more states.
Coinfection is a different concept
Sequential infection means one strain is followed by another at a later time. Coinfection means two strains infect the same host at the same time or with overlapping infection periods.
If coinfection matters biologically, an additional state such as \(I_{12}\) may be required. It should not be added automatically: extra compartments are useful only when the biological process matters to the modelling question.
Immune escape
A new strain may spread not only because it transmits faster but because existing immunity protects less effectively against it. This is called immune escape.
These mechanisms should be distinguished:
| Mechanism | What changes? |
|---|---|
| greater transmissibility | more transmission under otherwise comparable conditions |
| immune escape | previously immune people become more susceptible to the new strain |
| longer infectious period | infectious individuals have more time to transmit |
Two strains can therefore have the same intrinsic transmissibility but different growth in a partially immune population.
Vaccination in a multi-strain model
Vaccine protection may differ by strain. If a vaccine strongly protects against strain 1 but less strongly against strain 2, the vaccinated population does not present the same susceptible environment to both strains.
A model can therefore use strain-specific vaccine efficacies, for example \(e_1\) and \(e_2\), rather than one universal efficacy.
Strain-specific reproduction numbers
Each strain can have its own basic reproduction number in a fully susceptible population. In a very simple SIR-type setting,
\[R_{0,1}=\frac{\beta_1}{\gamma_1},\qquad R_{0,2}=\frac{\beta_2}{\gamma_2}.\]But once immunity is already present, the more relevant quantities are effective reproduction numbers, because each strain sees a different amount of effective susceptibility.
Thus a strain may grow when
\[R_{e,k}>1\]and decline when
\[R_{e,k}<1.\]Why two strains can coexist
A higher-transmission strain does not always eliminate every other strain. Coexistence can occur when strains occupy different ecological or immunological niches, cross-immunity is incomplete, immunity wanes, strains are repeatedly introduced, or their advantages vary over time or between host groups.
Multi-strain dynamics are therefore determined by both differences between strains and interactions between strains.
Mutation and the appearance of a new strain
A multi-strain model can begin with two strains already present, or introduce a second strain later. The appearance of a new strain may be represented as an external introduction or, in more detailed evolutionary models, through mutation.
For many epidemiological questions it is not necessary to model the molecular mutation process itself. It may be sufficient to specify when a new strain enters and then study whether it grows, declines or replaces existing strains.
What the data can tell us
Strain-specific case counts or genomic surveillance can be used to estimate the changing proportion of infections caused by each strain. A rapidly increasing strain frequency can indicate a transmission advantage, immune escape, or a combination of mechanisms; additional data are needed to distinguish them.
Main modelling assumptions
| Assumption | Question to ask |
|---|---|
| two discrete strains | is this enough to represent the relevant pathogen diversity? |
| fixed strain parameters | do transmissibility or immune escape change over time? |
| specified cross-immunity | is protection complete, partial, asymmetric or waning? |
| no coinfection | is simultaneous infection biologically important? |
| homogeneous population | do age, contact patterns or vaccination create important structure? |