We modelled the co-circulation of three variants. Inspired by [
21], we proposed a status-based multi-variant compartmental model allowing us to evaluate the delta, omicron and residual beta/gamma variants simultaneously, which interact with each other using a cross-immunity term. A schematic overview of the model is shown in Fig.
1A. As outlined in this figure the main advantage of this model is that allows us to simulate multiple variants simultaneously taking the immunity acquired, due to the infection by one variant against an other variant into account. This property is modeled using a cross-immunity matrix
\(\epsilon _{ik}\). The authors of [
21] have proposed a similar parsimonious process to model a large number of competing strains, with application to the influenza dynamics. We largely adapted this model in our study for the purpose of describing the dynamics of competing SARS-CoV-2 variants. The model is defined by the ordinary differential eqs. (
1–
4), where state variables stand for proportions of different compartments in the population from the viewpoint of each variant
i:
$$\begin{aligned} dS_{i}= \,& {} \left( \eta _{i}R_{i}-S_{i}\sum _{k=1}^n \epsilon _{ik}\beta _{k}I_{k} \right) dt, \end{aligned}$$
(1)
$$\begin{aligned} dE_{i}= \,& {} \left( \beta _{i}S_{i}I_{i}-\delta _i E_{i}\right) dt, \end{aligned}$$
(2)
$$\begin{aligned} dI_{i}= \,& {} (\delta _i E_{i} - \gamma _i I_{i})dt, \end{aligned}$$
(3)
$$\begin{aligned} R_{i}= \,& {} 1-S_{i}-E_{i}-I_{i}. \end{aligned}$$
(4)
\(S_i\) represents the population susceptible,
\(E_i\) the incubating non infectious population and
\(I_i\) the population of infectious individuals. Compartment
\(R_i\) models an immunized population that either underwent infection and recovered from the disease or has been vaccinated.
\(\beta _i\) represents the transmission rate,
\(\eta _i\) is the immunity waning rate,
\(\delta _i\) the rate at which exposed individuals become infectious, or the inverse of the mean sojourn time in
E compartment, and
\(\gamma _i\) the recovery rate or the inverse of the infectiousness duration. Variant interaction is modelled by a cross-immunity matrix, where element
\(\epsilon _{ik}\) describes the acquired protection to an acquisition of variant
i conferred by an infection with variant
k. For a given reproduction rate at time
t,
\(\hat{R}_k(t)\), the transmission rate
\(\beta _{k}\) can be obtained from:
$$\begin{aligned} \beta _{k} = \frac{\hat{R}_k(t=0)\gamma _k}{1-a_{k}}, \end{aligned}$$
(5)
where
\(a_k\) represents the immunization level in the population against variant
k at the beginning of the study period. In further detail, in eq. (
1) the
\(S_{i}\sum _{k=1}^n \epsilon _{ik}\beta _{k}I_{k}\) term models the cross-immunity and, at the same time, the exit from the susceptible compartment of newly infected individuals. It is therefore the most significant part of this model. Aforementioned
\(\beta _{k}\) as outlined in eq. (
1), modulates the strength of the infection, while the
\(\epsilon _{ik}\) matrix makes sure that variants
i are also affected by a pull resulting from infected individuals
\(I_{k}\), and takes, as such, care about the cross-immunity. Further equations of the model (
2,
3,
4) follow standard SEIR modelling procedures, with the main difference being that index
i accounts for multiple variants in coevolution. In the case of the omicron variant, the immunized fraction
\(a_{\textrm{omicron}}\) is obtained by multiplying the fraction of population immune against the delta variant
\(a_{\textrm{delta}}\) with the cross-immunity between omicron and delta:
$$\begin{aligned} a_{\textrm{omicron}} = a_{\textrm{delta}}\epsilon _{\mathrm {[delta,omicron]}}. \nonumber \end{aligned}$$
(6)
We assume that
\(\hat{R}_k(t)\) is constant over the investigated period, and we define the relative epidemic fitness of variant
i to the delta variant as the ratio of reproduction rates:
\(\frac{\hat{R}_i}{\hat{R}_{\textrm{delta}}}\).
Parameter values were either (i) based on literature values in the case of
\(\delta _i\) and
\(\gamma _i\), or (ii) hypothesised for
\(a_k\),
\(\eta _{i}\) and
\(\epsilon _{ik}\), with different values tested for robustness purposes for both (i) and (ii) (see Sect.
2.4), or (iii) estimated from data for
\(\beta _i\) [related to
\(\hat{R}_k\), equation (
5)]. In the baseline scenario, the mean generation time, which is expressed in our model for a variant
i as
\(1/\delta _i + 1/\gamma _i\), was assumed equal to 5 days for delta, 3.5 days for omicron [
23] and 8 days for the other variants (beta/gamma) [
24,
25]. The corresponding durations (
\(1/\delta _i\) and
\(1/\gamma _i\)) in
\(E_i\) and
\(I_i\) compartments were assumed equal to (2,3) days, (1.4, 2.1) days and (5,3) days for delta, omicron and beta/gamma variants, respectively. A further in detail overview of all rate constants evaluated is outlined in Table
1.
Table 1
Values used in the sensitivity analysis for mean generation time and duration in
E and
I compartments for each variant, from [
23‐
25]. All values are given in days
Mean generation time | 5 | 3 |
Mean duration in E \(1/\delta\) | 2 | 1.2 |
Mean infectiousness duration \(1/\gamma\) | 3 | 1.8 |
Mean generation time | 5 | 3.5 |
Mean duration in E \(1/\delta\) | 2 | 1.4 |
Mean infectiousness duration \(1/\gamma\) | 3 | 2.1 |
Mean generation time | 5 | 4 |
Mean duration in E \(1/\delta\) | 2 | 1.6 |
Mean infectiousness duration \(1/\gamma\) | 3 | 2.4 |
Mean generation time | 5 | 6 |
Mean duration in E \(1/\delta\) | 2 | 2 |
Mean infectiousness duration \(1/\gamma\) | 3 | 4 |
Mean immunity duration \(1/\eta\) | 1000 | 1000 |