key: cord-1022243-e9rlsaww
authors: Massard, Mathilde; Eftimie, Raluca; Perasso, Antoine; Saussereau, Bruno
title: A multi-strain epidemic model for COVID-19 with infected and asymptomatic cases: application to French data
date: 2022-05-02
journal: J Theor Biol
DOI: 10.1016/j.jtbi.2022.111117
sha: 80c5298fd6f1eb3253ec3cb6232c09a52f143a03
doc_id: 1022243
cord_uid: e9rlsaww

Many SARS-CoV-2 variants have appeared over the last months, and many more will continue to appear. Understanding the competition between these different variants could help make future predictions on the evolution of epidemics. In this study we use a mathematical model to investigate the impact of three different SARS-CoV-2 variants on the spread of COVID-19 across France, between January-May 2021 (before vaccination was extended to the full population). To this end, we use the data from Geodes (produced by Public Health France) and a particle swarm optimisation algorithm, to estimate the model parameters and further calculate a value for the basic reproduction number [Formula: see text]. Sensitivity and uncertainty analysis is then used to better understand the impact of estimated parameter values on the number of infections leading to both symptomatic and asymptomatic individuals. The results confirmed that, as expected, the alpha, beta and gamma variants are more transmissible than the original viral strain. In addition, the sensitivity results showed that the beta/gamma variants could have lead to a larger number of infections in France (of both symptomatic and asymptomatic people).

Since the beginning of 2020, a pandemic of a novel coronavirus disease has spread throughout the world and caused millions of deaths. COVID-19 is an infectious respiratory disease caused by the SARS-CoV-2 virus. The World Health Organization (WHO) learned of the existence of the SARS-CoV-2 on December 31, 2019 when an outbreak of "viral pneumonia" cases was notified in Wuhan, People's Republic of China [39, 34] . This disease has then spread to the whole world and very quickly led to saturated hospitals.

The fast spread of SARS-CoV-2 also led to the emergence of different variants (i.e., viral genomes that may contain one or more mutations) for this virus, who have been circulating around the world since the beginning of this pandemic. While most of the genetic mutations observed in the circulating SARS-CoV-2 variants do not significantly change virus biology and its properties, some fitness-enhancing mutations (e.g., mutations leading to an increase in transmissibility, or in ability to evade the immune response) have been observed after the first months of virus spread [19] . The circulating variants are classified as: Variants Being Monitored (VBM) -those for which data indicate a potential impact on approved treatments but might not pose yet a significant and imminent risk to public health, Variants of Interest (VOI)for which there is predicted increase in transmissibility of disease severity, and reduced efficacy of treatments, Variants of Concern (VOC) -for which there is evidence of an increase in transmissibility, increase in the severity of disease, and reduced effectiveness of treatments [14, 15] .

Variants can be re-classified based in their attributes and prevalence. As of October 2021, the European Centre for Disease Prevention and Control (ECDC) has listed three VOC (Beta, Gamma and Delta) and two VOI (Mu and Lambda) [15] .

To investigate the spread of COVID-19, including the role of the different SARS-CoV-2 variants on this spread, and to propose measures to slow-down this spread, researchers have focused their attention on various mathematical compartmental models, either stochastic [21, 12] or deterministic [9, 20, 33, 6, 38, 5, 18, 23, 32, 37] ; see also references therein. The great majority of these models focus on one single variant and investigate, for example, the impact of hospitalisation and quarantine [20, 33] , the impact of vaccination [38] , or the impact of different age classes [6, 9] . Over the last few months, when it became clear the importance of different VOC on the fast increase in the number of COVID-19 cases in different countries, a series of mathematical models have been derived to investigate the role of multiple variants on the spread of SARS-CoV-2 [5, 18, 23, 32, 37] . For example, Sridhar et al. [32] and Yagan et al. [37] used network models to investigate the effectiveness of mask-wearing in limiting the spread of COVID-19 in the context of viral mutations. Arruda et al. [5] considered a SEIR model that incorporated several viral strains and also reinfection due to vaning immunity. They studied time-varying control strategies in the context of lockdown measures, and so they focused on the cost of infection control (i.e., lockdown) vs. the cost of elevated infection levels to the healthcare system, over a two-year time interval. Khyar and Allali [23] focused on the global stability analysis of a two-strain SEIR epidemic model with two general incidence rates. They also calculated the basic reproduction number for their epidemic model. Finally, Gonzalez-Parra et al. [18] extended a two-strain SEIR model to include also asymptomatic, hospitalized and dead individuals, to investigate the transmission of COVID-19 in Columbia. They consider variations in the contagiousness of the two strains, to see their impacts on the number of infections, hospitalisations and deaths.

In this study we also focus on the role of different SARS-CoV-2 variants on the transmission of this virus across France in the absence of vaccination. However, unlike the previous studies, here we consider a simpler compartmental model (i.e., a generalisation of a SIR model) that includes also asymptomatic cases and dead individuals infected with the original variant, as well as cases infected with other variants. To calculate the basic reproductive number R 0 , we first focus on a general model with N > 0 variants, and derive a formula for R 0 . Then, we apply this formula for the case with two variants (N = 2), corresponding to the alpha and beta/gamma variants (see discussion on French data in Section 2). This model is parametrised (with the help of a particle swarm optimisation algorithm) using French data selected over the longest possible time interval that can take into account the emergence of different SARS-CoV-2 variants without including also the vaccination. Sensitivity analysis is performed to understand the impact of uncertainty in model parameters on the overall outcome (i.e., symptomatic and asymptomatic infections). This sensitivity analysis allows us to investigate the severity of infections with different variants in France, in the absence of any vaccination.

The paper is structured as follows. In Section 2 we introduce a general compartment model that considers a generic number of N variants, that can lead to symptomatic as well as asymptomatic cases. We use this model to calculate a general formula for R 0 in terms of the parameters associated with different variants. In Section 3 we discuss SARS-CoV-2 data for France, and based on this data we focus on a simpler model with N = 2 variants: alpha and beta.

Using available data, we parametrise this simpler model with the help of a particle swarm optimisation algorithm. In Section 4 we estimate the basic reproductive number for the three SARS-CoV-2 variants considered here, and we perform a sensitivity analysis to investigate the impact of uncertainties in the previously-identified model parameters on the model outcome (i.e., the number of infectious individuals, as well as R 0 ).

We start by presenting a general compartmental model that takes into account a generic number N of variants, with N > 0. We choose N variants because it allows us to include all current and possible future VOC variants. Symptomatic and asymptomatic cases are taken into consideration with the assumption that asymptomatic people do not die from the disease.

Natural births and deaths can be overlooked, due to the large size of the French population.

Loss of immunity can also be ignored because the number of individuals infected twice is still low [8] , and this loss of immunity is not present for all variants. For example, variants with the E484K mutation in the spike protein may be responsible for immune escape [28, 35] .

For the model considered in this study, we denote the initial strain by V 0 , the first variant by V 1 , ..., the N -th variant by V N . The population is divided into 2N + 5 different classes: sus- The dynamics of the population is described by the system below (see also Figure 1 for a schematic description of the interactions between different model variables). a A V 0 . In addition, individuals can leave this asymptomatic class following recovery at a rate +b V 0 . We remind the reader that the asymptomatic persons cannot die. Similar terms can be found in the equations describing the evolution of the other compartments A V 1 ,...,A V N ; see also Table 1 .

The equation for dR dt in (1) describes the evolution of recovered individuals. The terms that appear in this equation describe the recovery of infected and asymptomatic individuals, as discussed above.

The equation for dD dt in (1) describes the time-evolution of COVID-19-dead individuals.

The terms in this equation are the death terms that we discussed above for the symptomatic infected compartments.

The equation for dS dt in (1) describes the time-evolution of the susceptible people. These individuals leave the compartment S when becoming infected (after contact with symptomatic or asymptomatic people).

All parameters involved in this general model (1) with N viral variants are summarized in the Table 1 . In the following we discuss briefly the calculation of the basic reproduction number R 0 for this general model.

Mortality rate due to the initial virus and to other N variant

Transmission rate of variants and initial virus from symptomatic people

Transmission rates of variants and initial virus from asymptomatic people size −1 × day −1 

with N the size of the population considered (hence the units for these parameters are 1/(day × size)). Here, j k i and j k a are the average duration times between two contaminating contacts (with the strain k) by a symptomatic person and by an asymptomatic person.

A fundamental tool in epidemiology is the basic reproduction number R 0 , which represents the number of secondary infections resulting from a single infectious individual introduced into a fully susceptible population [29, 10, 30] . Note that in general a population is not fully susceptible, with some individuals already having immunity/cross-immunity to the virus. Therefore, even if throughout this study we refer to the basic reproductive number R 0 , we actually understand the effective reproduction number (i.e., the number of secondary infections resulting from a single infectious individual introduced into a population formed of susceptible and non-susceptible individuals); this will be more clear in Section 3 in the context of alpha and beta variants for COVID-19.

One method that can be used to calculate the basic reproduction number for the finitedimensional system (1) makes use of the next generation matrix [10] . This method is based on the definition of R 0 as the dominant eigenvalue of the "next generation matrix" [29] , i.e., a matrix that relates the numbers of newly infected individuals in various categories in consecutive generations [11] . Applying this method to model (1) we obtain the next generation matrix M below:

Thus, we obtain the following formula for R 0 :

Each eigenvalue consists only of the parameters corresponding to a single strain. It is like there are several epidemics at the same time, and they don't interfere with each other. By obtaining a value of R 0 from the available data (see Section 3), we will be able to conclude which variant takes over the others.

Throughout the rest of this paper we focus on a simpler model with only two variants, (i.e., N = 2), and we use the available data to obtain an estimate for R 0 (or rather the effective reproduction number, as discussed above). We estimate the 18 parameters involved in model (1) and in equation (2) using a Particle Swarm Optimization algorithm.

The data we use in this study is from Geodes (i.e., the cartographic observatory of epidemiological indicators produced by Public Health France 1 ), and covers the period between 12th

February and 7th May 2021, because before the 12 February the different strains of the virus were not recorded (only virus presence was recorded). Moreover, we consider data only until 7th May to avoid the effects of vaccines (since in France the vaccination really started off in May). We focus on the percentages of positive RT-PCR tests identifying the initial strain, the alpha variant (which appeared in France since mid-December 2020) and the beta & gamma variants (present in France since end of December 2020 & beginning of February 2021, respectively), calculated over a 7-day period. This period allows to remove the effect of reduced testing on Sundays. The data can be seen in Tables 4, 5 (1) (with N = 2) count the current numbers of infected individuals and not the newly infected per day. To transform the data so that we can use it to parametrise our model, we assume that individuals stay an average of 13 days in a symptomatic compartment before recovering [17] .

By including this recovery delay, we reduced the size of the three datasets from 85 to 73 days:

with I new denoting the rescaled cumulative data that consider the assumption of 13 days infections (i.e., we sum up all daily infections I original data that took place over the past 13 days), which is then used to parametrise the mathematical model.

The evolution of the three infected compartments can be seen in Figure 2 Data for initial population sizes. For the numerical simulations performed throughout the rest of the paper, we consider the following initial population sizes:

• Using the latest census data from mainland France [1] , in this study we assume that the total population of France is ≈ 64000000. We consider this initial condition because on the infected people, we assume that 85% are symptomatic and 15% are asymptomatic [36] , and that the asymptomatic are not tested the same way as the symptomatic infected individuals and thus we take 

The Particle Swarm Optimization (PSO) algorithm was created in 1995 by Russel Eberhart, electrical engineer, and James Kennedy, social psychologist [22] , and it is inspired by the collective behaviour of flocks of birds or schools of fish. For the PSO, the particle (i.e., potential solution of the model) moves by making allowances between getting closer to the optimal solution visited and getting closer to the solutions found in its neighborhood. This optimization algorithm performs well on parameter optimization for ordinary differential equation models [4] , but convergence to the overall optimal solution is not always guaranteed. In the following we describe how we apply this PSO algorithm to identify the parameters of our model (1) with two SARS-CoV-2 variants, using the data described above.

For each viral strain we calculate the residual sum of squares

• y k (t i ) describe the empirical observations at time t i corresponding to the number of individuals infected by strain k ∈ {V 0 , V 1 , V 2 }.

•ŷ k (t i ) describe the numerical predictions at time t i for the individuals infected by strain k ∈ {V 0 , V 1 , V 2 } (as given by system (1) with N = 2).

The objective function to be minimized is therefore

Based on reasonable parameter bounds (see discussion below), we constrain our search space This means that infected people take at most 25 days and at least 1 day to infect a person [27] . Indeed, the transmission rate has been estimated at 5.0981280×10 −12 for the whole world [26] which corresponds to 1/(7, 870, 000, 000 × 5.0981280 × 10 −12 ) = 24.9 days to transmit the virus (because N=7,870,000,000 is the world population). The symptomatic proportion is between 0.55 and 0.9 [36] , a symptomatically-infected person takes between 3 days [31] and 13

days [17] to recover, and between 5 days [13] and 13 days to die (also to match our infection hypothesis [17] ). In Figure 3 we show the best fit between the data collected from Geodes [2] on infected individuals (black dots), and the numerically-predicted number (red curve) of individuals infected with (a) initial strain, (b) alpha strain, (c) beta/gamma strain. The parameters for which the numerical solutions were obtained are summarised in Table 2, while Table 3 gives the biological interpretation of these parameters. The parameters α Table 1 .

The fit of the individuals infected with the initial and alpha strain is relatively good. In contrast, the fit of individuals infected with the beta strain is quite poor for time t ∈ [0, 10] days and t ∈ [40, 60] days.

The evolution of asymptomatic infected people is presented in the Figure 4 . We see that the number of asymptomatic individuals infected with the initial strain increases at the be-ginning and then decreases (following the trend of symptomatic infected individuals). The asymptomatic individuals infected with the other two variants (alpha and beta) follow the same general pattern as the symptomatic individuals but with fewer cases.

Value Symbol Value Symbol Value

1.353279e-08 Looking at the numbers in Tables 3 and 2 we see that the people who die the fastest are those affected by the alpha variant. Indeed, µ V 1 >µ V 2 > µ V 0 . (Note that this result is also supported by the values of the case fatality rate (CFR) for each of the three variants: CFR 0 = 5.94%; CFR 1 = 9.48%; CFR 2 = 7.98%.) In addition, people who take the least time to transmit the virus are those affected by the alpha variant which is consistent with the fact that it is said to be more transmissible. There are a higher number of days for the transmission of the disease to symptomatic people than for the asymptomatic people, and this is true for all three cases.

To calculate the value of R 0 , we have computed three non-zero eigenvalues for the next generation matrix: λ 0 = 0.898367 (corresponding to the initial strain), λ 1 = 1.141363 (corresponding to the alpha strain), λ 2 = 1.140814 (corresponding to the beta/gamma strain). It is clear that the two variants (with R 0 = 1.14) are more contaminating than the initial strain (with R 0 = 0.898). Both alpha and beta variants have very close values, so we can conclude at first that they have a similar intensity, at least for France, in the absence of vaccination. Note that this R 0 value is consistent with the value calculated by other French studies that investigated COVID-19 transmission during the same period of time [16] . 

We now investigate the impact on R 0 when we vary by one day the biological parameter estimates given in Table 3 . For example, if we estimated at 7.11 the number of days required for the transmission of the disease by a symptomatic person, we now consider the interval [6.11, 8.11] days. Each time we vary one parameter, while fixing all other parameters. The results are presented in Figure 5 . First, we see in Figure 5 (a) that the parameters of the initial strain do not impact R 0 because the corresponding value is never the maximum among the three eigenvalues of the next generation matrix. For the alpha variant ( Figure 5(b) ), we see that R 0 is especially sensitive to α V 1 i and α V 1 a , parameters describing the transmission of infection following contacts with symptomatic and asymptomatic infected individuals. Thus, reducing the contacts (e.g., via lockdown) could lead to a reduction in R 0 . Also notice that none of the parameters related to the alpha variant increase R 0 above 2. For the beta/gamma variants ( Figure 5(c) ), we see that R 0 is very sensitive to α V 2 a . Changes in this parameter can increase R 0 up to R 0 ≈ 6 (unlike the case for the alpha variant, where R 0 was always below 2). This shows us the importance of controlling the α V 2 a parameter to be able to control the outbreak of the beta/gamma variant. Another parameter that impacts R 0 is b V 2 , the recovery rate of asymptomatic individuals. This shows the key role of the asymptomatic people in the rapid spread of the beta/gamma strain.

Since knowledge about model parameters is incomplete (especially knowledge about the role of asymptomatic people on disease transmission), we are now performing a global sensitivity and uncertainty analysis of the different infected compartments, where we vary all parameters at the same time. To this end, we consider a classical approach that combines the Latin Hypercube Sampling (LHS) with the Partial Rank Correlation Coefficient (PRCC) [25] .

The parameters are varied in the same way as before (i.e., by one day for each of the rates), and the sample size for the LHS is 100. Figure 6 : Uncertainty in the I V 0 (t) and A V 0 (t), as we vary by a maximum of ±1 day all model parameters (according to the LHS scheme).

In Figure 6 we show the variation in the number of individuals infected with the original strain, as we vary all model parameters. The black curves show the median time evolution of I V 0 (t) and A V 0 (t), while the gray regions show the range between quantiles q25 − q75 and q5 − q95. We can see that both I V 0 and A V 0 are very sensitive for t ∈ [10, 30] days. For example, a 1-day change in the various transition rates incorporated into the model, leads on day t = 10 (i.e. February 22) to a variation between 50,000 to 80,000 in the symptomatic infected people (I V 0 ), and a variation between 10,000 to 25,000 asymptomatic infected people approximately. In Figure 7 we show the variation in the number of individuals infected with the alpha strain, as we vary all model parameters. Note here the change in the maximum peak of both I V 1 and A V 1 : not only the amplitude but also the shift in time: from day t ≈ 9 for the median black curve, to day t ≈ 13 for the q5−q95 quantiles. If we compare these results with the results in Figure 8 for the beta/gamma strain, we observe that while the maximum for I V 1 is reached around t = 13 days, for I V 2 the maximum is reached for t ≈ 9 days. Therefore, the beta/gamma strains are more aggressive. Moreover, by comparing Figures 7-8 , we see that I V 2 and A V 2 could reach much higher amplitudes (i.e., 10-fold higher) compared to I V 1 and A V 1 . We will return to this aspect in the Discussion section 5.

Finally, for completeness, in Figure 9 we also show the time-variation in the recovered R(t) and dead D(t) people. Again, we see a large variability in the outcome between days t = 10 and t = 30 which, for D(t) lasts also for t > 30 days.

To quantify the sensitivity of model outcomes (i.e., number of symptomatic and asymptomatic infected individuals) to changes in model parameters, and identify the critical parameters, in Figures 10-12 we show the Partial Rank Correlation Coefficient (PRCC) -a samplingbased measure for nonlinear but monotonic relationships between inputs and outputs [25] .

We see here that the parameters with the greatest impact on the outcome (i.e., parameters with PRCC indexes above 0.5 and below −0.5) are:

• For I V 2 : parameter p V 2 and α V 2 a ; while for A V 2 : parameters b V 2 and p V 2 .

First we observe that due to the interactions between the model variables, parameters associated with the second variant (V 2 ) impact the infections with the original (V 0 ) strain and the infections with the first variant (V 1 ). For example, p V 2 impacts not only I V 2 and A V 2 , but also I V 0 , A V 0 , I V 1 and A V 1 , which suggest that the people infected by beta/gamma strain could have a strong impact on the overall evolution of epidemics. This is despite the fact that for R 0 , the impact of beta/gamma variant was decoupled from the impact of the original variant and alpha variant (see Eqn. (2)).

In this paper we investigated the evolution of COVID-19 epidemic in France between February-May 2021 in the presence of multiple viral strains, while estimating the parameters that char- were considered together here, because in France the tests did not discriminate between them.

First, we showed that the value for R 0 was given by the maximum of three basic reproduction values corresponding to the three different variants. This suggested a sort of decoupling between the evolution of different variants, the most transmissible one having the greatest influence on the evolution of the outbreak. Using French data on Geodes [2] and a PSO algorithm, we estimated R 0 = 1.14, consistent with the value estimated by other French studies [16] . Moreover, this R 0 value was similar for the alpha and beta/gamma variants, but larger than the value corresponding to the original strain (thus suggesting that between February-May the alpha and beta/gamma variants were more transmissible than the original strain). A local sensitivity analysis for R 0 showed that the parameters with the largest impact were the transmission rates of symptomatic and asymptomatic individuals; see Figure 5 .

In regard to parameter estimation (see Table 3 ) we also found a higher number of days necessary to create contaminating contacts for the symptomatic people than for the asymptomatic people (for all three variants). This unexpected result could be explained by the fact that asymptomatic people come into contact with more peers ignoring distancing rules. The results in Table 3 show also that the number of days to recover when symptomatic is slightly lower than for asymptomatic people for the alpha variant, while it is almost twice more for the other strains. It would be interesting to investigate whether there is a specificity of the alpha variant in contrast to the other variants. At this point we could not find any epidemiological data to test the validity of this theoretical result.

Second, a global sensitivity and uncertainty analysis identified the variations in the ampli- tude of symptomatic and asymptomatic infections for all three strains, as well as the day when these infections peak (see Figures 6, 7 and 8) . Moreover, the PRCC analysis identified some critical parameters for the evolution of the epidemics (see Figures 10,11 and 12) . The interesting result was that parameters associated with the V 2 variant (p V 2 ) seemed to impact also the evolution of infections with the V 0 and V 1 variants, thus suggesting that the infections with the different viral strains are not really decoupled (as suggested by the R 0 formula (2)).

The uncertainty analysis also showed that the evolution of those infected with the beta/gamma strain i.e. V 2 has more uncertainty compared to the other two strains (when we vary the interaction rates by one day).

Also in regard to the uncertainty of the results in this study, we need to mention the fact that since we could not find mortality data for each of the different strains discussed here, the exact identification of parameters µ V i actually depends on all other model parameters listed in Table 2 .

Throughout this study we chose the average infection period to be 13 days, since this was similar to other published studies [17] . Even reducing this period by 1-2 days did not seem to lead to significant changes. However, in the future it would be interesting to investigate model dynamics when we change (increase/decrease) this infection period in a more significant way.

This study focused on the dynamics of COVID-19 epidemics in the presence of alpha and beta/gamma strains, with distancing measures (and masks worn indoor and outdoor) but no vaccinations (as the vaccination program in France really took off in May 2021). Our results about the potential large amplitude in the infections with beta and gamma strains, can be un-derstood in this context: these strains would have caused a disastrous outbreak situation if the situation had not changed with the arrival of the vaccine. The subsequent emergence of the delta variant also changed the dynamics of the epidemics, and if we include it into our model (1) the simulation results and predictions will inevitably change.

The study can be further extended to the new SARS-CoV-2 variants that are appearing, which might avoid the anti-viral immune responses generated by the vaccination. In regard to this, a very recent review [24] discussed vaccine effectiveness in the context of the multiple SARS-CoV-2 variants that emerged over the last months, and concluded that since a large number of variants have mutations mainly associated with the spike protein, which is also a key component of most of the vaccines on the market, vaccine efficacy needs to be assessed for each variant. While some vaccine efficacy studies have been published for earlier variants [7] , many more such studies are ongoing.

In Tables 4-8 We note that the values in Table 7 , corresponding to the number of cases on Mondays, are lower because fewer tests are carried out on Sundays and therefore fewer new cases are confirmed the next day. To remove this effect, we take an average over 7 days at each time, and obtain the data summarised in Table 8 . 

Table 4: Percentage of initial strain

Table 5: Percentage of alpha variant

Table 6: Percentage of beta variant

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