CC BY
10Russian Journal of Nonlinear Dynamics, 2021, vol. 17, no. 4, pp. 475-490. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd210409
NONLINEAR ENGINEERING AND ROBOTICS
MSC 2010: 34A34, 37M05, 37N25, 39A30, 92D30
Modelling the Effect of Virulent Variants with SIR
G. Nakamura, S. Plaszczynski, B. Grammaticos, M. Badoual
We study the effect of an emerging virus mutation on the evolution of an epidemic, inspired by the appearance of the delta variant of SARS-CoV-2. We show that if the new variant is markedly more infective than the existing ones the epidemic can resurge immediately. The vaccination of the population plays a crucial role in the evolution of the epidemic. When the older (and more vulnerable) layers of the population are protected, the new infections concern mainly younger people, resulting in fewer hospitalisations and a reduced stress on the health system. We study also the effects of vacations, partially effective vaccines and vaccination strategies based on epidemic-awareness. An important finding concerns vaccination deniers: their attitude may lead to a prolonged wave of epidemic and an increased number of hospital admissions.
Keywords: epidemic, vaccination, seasonality, recruitment, SIR model
1. Introduction
Mathematical models can be particularly useful when applied to the study of epidemics. The domain is such that any experimentation or result replication attempt is out of the question when the epidemic concerns the human population. A well conceived model can, through robust conclusions, guide policies aiming at the mitigation and/or confinement of the epidemic. Building an efficient model is always a tall order [1]. The first difficulty in the case of epidemic modelling is associated with long-term predictions. Epidemics having a major societal impact make it necessary for the governments to introduce measures which have as a consequence the modification of the model parameters making their predictions, at times, questionable. The second difficulty stems from the fact that the said parameters are not always accurately known. Thus, when one decides to enrich a model by adding more effects, this usually introduces more uncertainties
Received July 13, 2021 Accepted October 12, 2021
Gilberto Nakamura
nakamura@ijclab.in2p3.fr
Stéphane Plaszczynski
plaszczy@lal.in2p3.fr
Basil Grammaticos
grammaticos@imnc.in2p3.fr
Mathilde Badoual
badoual@ijclab.in2p3.fr
Université Paris-Saclay
CNRS/IN2P3, IJCLab, 91405 Orsay, France
Université de Paris
IJCLab, 91405 Orsay France
which may influence in a non-insignificant way the results. Faced with this difficulty, the present authors' choice is to keep a model as simple as possible, albeit without sacrificing realism, and thus use it to infer only the most salient conclusions.
Our work is motivated by the ongoing Covid-19 pandemic. Given the global disruption caused by the emergence of the SARS-CoV-2 virus, it is natural that major efforts would aim at the mitigation of the epidemic. The approach is two-pronged: treating infected individuals and preventing the spread of the epidemic. The first lies in the domain of medicine and pharmacology and thus somehow less amenable to mathematical modelling. Prevention, on the other hand, is where modelling can be most useful. Since the onset of the epidemic the main strategy of the governments and, to tell the truth, the only one initially available was that of non-pharmaceutical interventions (NPI), consisting in a series of restrictions and recommendations. The various NPI strategies have been the object of a substantial literature. N. Perra [2] published recently a review article containing a bibliography of close to 40 articles dealing with the matter. Several governments have set up panels monitoring the effect of NPI, but the general conclusion is that those interventions are not always effective [3]. This is due to the fact that epidemic-inducing diseases and human behaviour are intimately related and, in fact, in ways not clearly understood. The European Centre for Disease Prevention and Control in their report [4] points out that, while the NPI did succeed in reducing the impact of Covid-19 in Europe, they also had a negative impact on the general well-being of people, the functioning of society, and the economy in general. Their recommendation is that the NPI be tailored to the local situation, avoiding "one size fits all" type measures. Our work [5] adopted precisely such a point of view. In that work we analysed the first wave of the epidemic in France, which was accompanied by a lockdown mandate, and investigated the possible lockdown-exit strategies. Given that the relaxation of the confinement measures could be accompanied by a new epidemic wave, we investigated possible ways to moderate its amplitude. In a similar work Fokas and collaborators [6] studied the affect of lockdown measures in Greece, investigating the possibility of age-specific exit strategies.
While NPI can indeed contain an epidemic or at least limit its spreading, the best prevention strategy is vaccination. And, of course, it is not limited to the Covid-19 epidemic. The Vaccine Impact Modelling Consortium [7] investigated the impact of vaccination against a collection of pathogens concluding that millions of deaths can be avoided with timely vaccination. In the case of the Covid-19 pandemic the production, in record time, of efficient vaccines, moreover available in large numbers, was a real game-changer. The large-scale vaccination campaigns, first in Israel, then in the US and now in most European countries have established the efficiency of the vaccines. The data from Israel allowed Mahase [8] to conclude that "Israel sees new infections plummet following vaccination". A report for the US Center for Disease Control [9] followed 100 million vaccinated parsons registering around 10 thousand breakthrough infections, confirming the exceptional vaccine efficiency. In [10] we studied the effect of vaccinations on the number of hospital admissions, focusing on the possible delays on the launching of the vaccination campaign as well as the consequences of age-prioritisation.
What has motivated the present study is the appearance of a new virus mutation, the variant know today as "delta" [11]. First detected in India, the variant is characterised by a substantially enhanced infectivity [12], even when compared to the variant known as "alpha", first detected in the United Kingdom [13], which was largely responsible for the appearance of a third epidemic wave in a slew of countries. The appearance of the delta variant has raised several questions, among which the prominent one is "will there be a fourth wave", and if yes, "when will it arrive", "will it become the dominant variant" [14],"who will it mainly affect" [15], "what will be its impact on the health system" and so on. We intend to address these questions in the framework of a model based on the SIR one used in previous works of ours. Moreover, we shall investigate the effect of population circulation and concentration associated to the vacation period. Since the first assessments point towards a slightly diminished efficiency of the existing vaccines in dealing with the delta variant [16], we will include such an effect in our model. By going back to our two (young-old) population models, introduced in [10], we shall try to identify the sub-population
mainly concerned by the epidemic resurgence, the impact of the latter on hospitalisations and the possibility to mitigate the consequences by a targeted acceleration of the vaccination campaign. Finally, we shall study the consequences of the attitude of vaccination deniers of the evolution of the fourth wave.
2. The model
The continuous model
The model we shall use in the present study is the venerable SIR model. It was introduced a century ago by Kermack and Mc Kendrick [17] and, in its simple, original form, considers a population split into three parts: the "susceptible" individuals who can catch the infection S, the "infective" ones who can transmit the disease I, and the "removed" R who either died from the disease or, having recovered, are immune to it. The SIR model is expressed as a non-linear differential system
^ = -aSI, dt
^ = aSI-XI, (2.1)
dt
f = A/, dt
where a is the infection rate and A the removal rate of the infected individuals. The form of (2.1) ensures that the total population S +1 + R is a conserved quantity. Assuming that S, I and R are fractions of the population, we can normalise this quantity to 1. The ratio a/A defines what is called the basic reproduction number, usually referred to as R0. It corresponds to the number of infections in the susceptible population resulting from a single infection. The parameter A fixes the time scale. In the case of the COVID-19 epidemic the value of A_1 is 5 days, as we argued in [5]. Many modelling studies, aimed at the Covid-19 epidemic, favour the use of the SEIR model over that of SIR [18]. However, as argued in [19], we prefer the use of the simpler model. Our main argument, based also on the work of Aleta and Merano [20], is that there exists a perfect correlation between the results of the two models. The main effect of the presence of the extra, "exposed", component in SEIR is just a tunable delay in the growth of the epidemic compared to the SIR results.
When a new variant of the virus appears, with characteristics different from the existing one, and starts infecting people, the equations should be modified accordingly. We introduce a new infective population J and obtain the system
dS
— = -aSI - bSJ, dt
^ = aSI - XI, ddJt (2.2)
-j- = bSJ - A J, dt
dR
- = A(/ + J),
where b is the infection rate of the new variant. Since we assume that the new variant appears only at some point during the evolution of the epidemic, the quantity J is initially zero and starts growing when a small, seed, quantity is introduced at some time step. For simplicity we assume the same value of A for the two variants, accounting for any possible difference through the value of the infection rate.
After a year and a half of the epidemic a feature is emerging: it displays seasonal variations. Already at the appearance of the epidemic, laboratory studies [21, 22] of the effect of ultraviolet radiation on the virus had hinted at a possible seasonality. More studies are now supporting this argument based on an analysis of existing data [23]. Watanabe [24] compared the evolution of the epidemic over the first few months of the year in the north and south hemispheres obtaining substantial differences, providing thus clear statistical evidence in favour of seasonality. Introducing seasonality in the SIR model is straightforward: it suffices to make the parameter a time-dependent and assume that it varies periodically around some mean value with a period corresponding to one year [24]. A simple way to do this is by introducing the sinusoidal expression a(t) = a + / cos(2nt/T) where T stands for the duration of a year. The phase of the cosine chosen corresponds to the maximum of a(t) occurring at the beginning of the year, representing the situation in the northern hemisphere. A phase shift of T/2 would be necessary in the case of the southern hemisphere. All our simulations will deal with a north-hemisphere epidemic which exhibits seasonality, waning during winter, waxing during summer and reappearing again during the autumn period.
Starting from the simple SIR model, we shall now modify it (minimally) so as to take into account the various effects we wish to study, starting with vaccination. One important question that can be raised is that of immunity. Several studies have concluded that a Covid-19 infection generates a substantial immunological memory [25] and thus individuals who have recovered can mount an effective response upon re-exposure [26]. Concerning the immunity conferred by vaccination we believe that the extensive statistics of the CDC settle the question. Thus, we construct a model without re-infection. Assuming a constant-rate vaccination, we have
^ = —a(t)SI — v, dT
j^ = a{t)SI-\I, (2.3)
dR , r
■w = XI+v•
where v is the (constant) vaccination rate. This is not an essential constraint. We can easily make the rate time-dependent and also dependent on the other components of the model as explained in [10] where we simulated the effect of vaccination awareness. The tacit assumption in (2.3) is that the vaccine efficiency is perfect. However, in view of the first results concerning the delta variant, this may be particularly optimistic. Relinquishing this assumption does not present any problem. We assume that the vaccine has an efficiency f, smaller than 1 [27], and this results in a number of vaccinated individuals who can be infected and introduce the population U of persons who remain susceptible after having been vaccinated. (We can also assume that the fact that they are vaccinated attenuates somewhat the probability of their being infected, resulting in an infection rate c smaller than the initial one a. However, in the absence of hard evidence on this point we shall not allow for such a freedom in our simulations, taking c = a.) The SIR system for partially efficient vaccination is
= ~ v> r§ = vf- cm i,
dt (2.4)
— = a(t)(S + U)I-XI,
The SIR model, simple and realistic as it is, has a serious drawback: it lacks spatial dependence. The main assumption in (2.1) is that everybody is in contact with everybody else, as if they were
concentrated on a single point. This is not what is observed in the evolution of the epidemics, which have usually a rich spatial structure [28]. Travelling plays an essential role in the spreading of the epidemics. Several approaches [29, 30] have been contemplated, in the pre-Covid era, in order to palliate this SIR shortcoming. In [19] we introduced our own approach to population recruitment, which consists in assuming that there is an influx of susceptible individuals. Introducing the recruitment rate ¡(t) we obtain the system
dS
= -a(t)SI + Li.(t)(l - S),
dT
— = a(t)SI-XI-(i(t)I, (2.5)
dR ,T . .
w = x i-,.m.
Given the form of (2.4), it is clear that, if we start with S + T+R = 1, the recruitment effect does not modify this constant of motion. The simplest case, explored already in [19] (and which will be considered in this paper), is the vacation-associated recruitment. In this case ¡(t) is a step function, ¡ = 0 for t < 7 months, ¡ = ¡0 for 7 <t< 9 months, and ¡ = 0 for t > 9 months (for European countries).
Discretising the SIR model
In order to proceed to simulations, it is necessary to obtain a discrete form of the SIR equations. Instead of using a black-box integration routine, we prefer to construct our own discretisation based on the method we have presented in [31]. It follows the ideas of Mickens [32, 33] and is based on the fact that the quantities appearing in the SIR model are by definition positive. (In the case of SIR, Hethcote [34] has shown that the positivity of the solution is guaranteed provided one starts from positive initial conditions.) The prescription proposed in [31] can be cast in the simplest form: "if all quantities are positive, no minus sign should appear anywhere". The question of the stability of our discretisation approach was addressed in [35] and the adequacy of the method was once again assessed in [31].
We start from (2.3), introduce a forward difference of the time derivative, with time step 5, a staggering in the non-linear terms. The only remaining minus sign comes from the vaccination term. The same difficulty was encountered in [31] where we proposed a simple solution. It suffices to replace the —v term by -vS/S and introduce the staggering —vSn+\/Sn. We obtain thus the discrete analogue of (2.3)
S S S
■^n+l n _ T Q t ra+l
-- = anJnSn+1 ~~ AJn+1, (2.6)
5
Rfl+1 Rfl _ \ T , Sfl+l
5 -Xln+1+V~s7-
and, solving for the points at (n + 1), we have
o _ ^
S=
n
1 + anSIn + <j
I.
+ anàSn+1) (2.7)
ra+1 1 + X5
S
Sn
n
Given the structure of (2.7), it is clear that Sn + In + Rn is a conserved quantity and, since we shall work with fractions of the population, we normalise it to 1.
For completeness sake we give also the discrete equations corresponding to the population recruitment, as modelled by system (2.5). We start from the solution at point (n +1) obtained, say, from (2.7), and introduce the recruitment function M = Sy. We find
q __I ^ ^ II
n 1 + Mn '
4 = (2-8)
R= R=
1 + MJ R
1 + M„
It is clear that, thanks to the presence of the denominators, we have Sn + In + Rn = Sn + In + + Rn = 1. As already mentioned the estival habits in most European countries are such that a non-negligible part of the population travels to the preferential destinations of vacationers. Our recruitment function will be a simple constant M, non-zero over two summer months.
3. Simulation results
In this section we are going to present results obtained from simulations based on equation (2.3). In all these simulations we assume that the epidemic is seasonal, the amplitude of its infectivity being modulated by a factor 1 + 0.25cos(2nt/T) and we fix A_1 = 5 days. Moreover, unless otherwise explicitly stated, we assume that the vaccine efficiency is total. The first set of results concern a single population without any age stratification.
Results for a single component population
For all the figures presented in this section, the simulation starts with I(0) = 1 • 10_7, R(0) = 0 and S(0) = 1 — I(0) for the first variant. The second variant is introduced at time t0 = = 5 months and J(t0) = 1 • 10_6. When present, vaccination starts at t = 2 months. The reproduction number varies with time because of seasonality: a(t)/X = a0(1.0 + 0.25cos(2nt/T))/A for the first variant and b(t)/X = b0(1.0+0.25cos(2nt/T))/A for the new variant, and T = 12 months. The values of the reproduction numbers are given in the captions.
Figure 1 shows what would be the expected evolution of a moderately virulent epidemic under a low-rate vaccination campaign. Under the influence of seasonality the number of infections abates during the summer months only to increase again with the arrival of autumn. Still the effect of vaccinations (provided the rhythm is not disrupted) is such that the second, winter, wave is of lower intensity than the first.
Next, we assume that a slightly more infective variant of the virus makes its appearance at the end of May (at which point 25 % of the total population have acquired immunity through vaccination after having recovered from infection) and study its effect on the second wave under the assumption that the vaccination rate does not change. The results of the simulation are shown in Figure 2. We remark that the number of infections has increased substantially and that almost all are due to the new, more infective, variant which has essentially replaced the previous one. However, while the new variant becomes dominant, the temporal evolution of the epidemic does not change. The second wave arrives only in autumn.
In order to investigate the possible behaviour of an epidemic resurgence due to the delta SARS-CoV-2 variant, some further assumptions are in order. First, we assume that the new variant is substantially more infective than the already circulating ones. Second, we tailor our
Fig. 1. Temporal evolution of the infective fraction of the population under the action of the first variant, with seasonality and vaccination. The reproduction number is a0/X = 1.4. For vaccination, v/X = 5 • 10-4 and its efficiency is 100 % (f = 0)
0.040 T
Fig. 2. Temporal evolution of the infective fraction of the population, with seasonality and vaccination, for the two variants. For the first variant (black line), a0/X = 1.4. The delta variant (red line) is introduced at t = 5 months and is characterised by an infectivity b0/X = 1.8. The rate of vaccination is: v/X = 5 • 10-4 and its efficiency is of 100 % (f = 0)
conditions to those prevailing in France (and, in fact, in most European countries) where the vaccination rate is such that by the time the delta variant did appear (at the end of May), roughly 50 % of the population had already acquired immunity.
Figure 3 shows the results of such a simulation. The situation here is totally different from that depicted in Fig. 2. The new epidemic wave appears immediately, during the summer months. However, also due to the ongoing vaccinations, the amplitude of the wave is not excessively large and the wave disappears before the arrival of winter, when the quasi-totality of the population is vaccinated.
Fig. 3. Temporal evolution of the infective fraction of the population, with seasonality and vaccination, for the two variants. For the first variant (black line), a0/X = 1.4. The delta variant (red line) is introduced at t = 5 months and is characterized by an infectivity b0/X = 4.0. Vaccination starts at t = 2 months with a rate of v/X = 1.8 • 10~2 and an efficiency of 100 % (f = 0)
Fig. 4. Temporal evolution of the infective fraction of the population, with seasonality, vaccination and vacations, for the two variants. For the first variant (black line), a0/X = 1.4. The delta variant (red line) is introduced at t = 5 months and is characterized by an infectivity b0/X = 4.0. Vaccination starts at t = 2 months with a rate of v/X = 1.8 • 10~2 and an efficiency of 100% (f = 0). For the vacations n0/X = 5.0 • 10~4
At this point it is interesting to study the effect of disrupting factors, ones that could alter the details of the new-variant-induced wave. The first such effect is that of vacations. In previous publications of ours we have introduced the notion of recruitment, whereupon, due to the population circulation and concentration in the locations preferred by vacationers, the number of susceptible individuals temporarily increases. The details of the recruitment mechanism were
presented in [19] and we shall not go into them here. Suffice it to say that we use Eq. (2.5), or rather its discrete equivalent (2.8), with a recruitment during the months of July and August.
The results are shown in Fig. 4. We remark that the effect of vacations is far from devastating. The amplitude of the wave does increase but its temporal extension is not affected. Just as in the non-recruitment case the wave disappears by autumn.
The second disrupting factor we have studied is the effect of partially efficient vaccinations. The matter is not yet settled but the consensus is that the current vaccines are slightly less efficient when it comes to the delta variant. We have thus investigated this effect by assuming a 90 % efficiency.
Fig. 5. Temporal evolution of the infective fraction of the population, with seasonality and a partially inefficient vaccination against the second variant. For the first variant (black line), a0/X = 1.4. The delta variant (red line) is introduced at t = 5 months and is characterized by an infectivity b0/X = 4.0. Vaccination starts at t = 2 months with a rate of v/X = 1.8 • 10~2 and an efficiency of 90% (f = 0.1). For the vacations n0/X = 5.0 • 10~4
Our results are presented in Fig. 5. Just as in the case of the vacation effect, the slightly inefficient vaccines do not alter the epidemic dynamics. The wave is amplified, but does not extend beyond autumn. Despite their diminished efficiency, the vaccines manage to curb the epidemic. And, though we are not presenting a figure for this, the same conclusion holds when one combines the effects of vacations and slightly less efficient vaccine. The amplitude of the wave increases, but its temporal extension does not.
Next, we investigate the behaviour of the new-variant-induced epidemic wave on a population where we distinguish two age groups.
Results for a two-component population
Up to now we have assumed that the population constituted a single group with the same characteristics as far as the infection is concerned. However, it is clear that the various age groups do not respond in the same way to the infection risk and the requirement of hospitalisation, to say nothing of the probability of demise. In [10] we presented a SIR-based model where the population was split into two sub-groups. As explained there, the infection rate a is not constant anymore. Instead we have a matrix aij, which corresponds to the rate with which the sub-group j is infecting the sub-group i. We have argued in [10] that, unless one wishes to ask a very specific question, it is better to stick to the simplest possible generalisation of
the single-component population, namely, a two-component one. One group consists of the younger population up to roughly 60 years of age, the remaining comprising the older individuals. We decided that the "young" correspond to 70% of the total, while the "old" make up the remaining 30% (these proportions are reasonable for most European countries). The young-old split allows one to answer in a more realistic way questions about the number of hospital admissions and/or deaths. The assumption is that, given the age split we have opted for, the probability for hospitalisation is 5 times smaller for the "young" group as compared to that of the "old". As explained in [10], the number of hospitalisations in a sub-group is proportional to the total number of "removed" that have not been vaccinated and the total number of hospitalisations is the sum of the hospitalisations in the two sub-groups. The model we used is:
dSo ~dt dSy
~dt dit ~dt dIy
~dt
= — n S T — n S T — v
- oo o o - oy o y - o
= — n S T — n S T — v
- yo y o - yy y y - y
dIo
77" d,,nS..In XI.
oo o o yo y o - o
(3.1)
noy SoTy + nyySy Ty XIy,
dRo , T ,
= a In + vn,
dt dRv
-dr = XIv + «v>
where a and v may vary with time. Again, the total population is conserved and we can introduce the normalisation So + Io + Ro + Sy + Iy + Ry = 1.
The seasonality applies to each element of the matrix: for the first variant, aij(t) = aij(1.0 + + 0.25cos(2^t/T)), with T = 12 months and we define Rij(t) = aij(t)/X. We have performed simulations using the same parameters for the initial reproduction matrix as in [10]: Roo = 1.5, Ryo = 1.0, Ray = 1.0, Ryy = 1.2, where Ry represents the number of infections among the "old" from one infective "young" person. We multiply the reproduction numbers for the new variant by a global factor of 3, in order to account for its increased infectivity. For all the simulations Io(0) = 3.0 • 10"6 and Iy(0) = 7.0 • 10_6. The second variant always appears at t0 = 5 months, Jo(t0) = 1.0 • 10"6 and Jy(t0) = 1.0 • 10"6.
The vaccination rate was the same for the two groups, but the vaccinations of the young sub-group started one month later than those of the old sub-group. As a result, at the moment of the appearance of the new variant two thirds of the old group were vaccinated, while for the young group the proportion was not quite one fourth.
The results of the simulation are shown in Fig. 6. We remark that the epidemic extends now up to the onset of winter, but it affects almost exclusively the "young" population. Due to this fact the number of hospitalisations associated with this second wave are just 10 % of those of the first one. Moreover, most are related to "young" persons admissions, the old ones accounting for less than 20 % of the total. This is obviously due to the fact that the younger sub-group is not well protected by vaccination.
In order to investigate the effects of vaccination in the younger sub-population, we performed two different simulations. In the first we assumed that the vaccination campaign aimed at that group was intensified and a 30 % increase was introduced in the vaccination rate.
The effect of the targeted vaccination we observe is to limit both the amplitude and the temporal extension of the second epidemic wave, see Fig. 7. Moreover, an analysis of the results
Fig. 6. Temporal evolution of the infective fraction of the population, with seasonality and vaccination, for the two variants, in the case of two age groups in the population. The continuous line represents the younger sub-group, while the dashed line corresponds to the older population sub-group. Vaccination starts at t = 2 months for the older sub-group and at t = 3 months for the younger one with a rate of v/X = 1.1 • 10-2 for both sub-groups. The efficiency of the vaccination is 100 % (f = 0)
Fig. 7. Temporal evolution of the infective fraction of the population, with seasonality and vaccination, for the two variants, in the case of two age groups in the population. The continuous line represents the younger sub-group, while the dashed line corresponds to the older population sub-group. Vaccination starts at t = 2 months for the older sub-group and at t = 3 months for the younger one. The rate of vaccination is: v/X = 1.1 • 10~2 for the older sub-group and 1.43 • 10~2 for the younger one. At t = = 6 months, the rate of vaccination of the younger sub-group is multiplied by a factor of 1.3. The efficiency of the vaccination is 100 % (f = 0)
predicts a substantial reduction of hospitalisations which are now merely a few percent of those of the first wave.
In the second simulation we introduce the vaccination-awareness effect first proposed in [10]. We consider that the vaccination of the older sub-group pursues with constant rate, while for the younger sub-group we assume that the vaccination rate is directly related to the number of
infected individuals, increasing or decreasing with the number of infective. The effect of this awareness is slightly counter-intuitive. Clearly, during the summer months, when due to the seasonality the epidemic ebbs and the number of infective diminishes, there are relative few among the younger population who think about getting vaccinated. But, since during winter the epidemic results in a large number of infections in the younger population, motivating thus accrued vaccinations, the percentage of younger individuals who are vaccinated when the virulent variant appears is now close to 40%.
Fig. 8. Temporal evolution of the infective fraction of the population, with seasonality, and two variants, in the case of two age groups in the population and with a constant rate of vaccination for the older group and a variable rate of vaccination for the younger one (proportional to the number of infective). Again, the continuous line represents the younger sub-group, while the dashed line corresponds to the older population sub-group.Vaccination starts at t = 2 months for the older sub-group and at t = 3 months for the younger one. The rate of vaccination is v/X = 1.1 • 10-2 for the older sub-group. For the younger one, the rate of vaccination is proportional to the density of infective (with a factor of proportionality equal to 2.0 • 103). The efficiency of the vaccination is 100 % (f = 0)
The results of the simulation are shown in Fig. 8. Clearly, the behaviour of the epidemic is somewhat intriguing. The wave, instead of reaching its apex during the summer months, is delayed towards autumn. Again it concerns almost exclusively the younger population. As a consequence, the number of hospitalisations does not soar, as one could fear, looking at the number of infections, and does not exceed 25 % of the ones registered during the first wave.
4. Discussion
This paper had a double motivation. On the one hand, the appearance of the variant delta of SARS-CoV-2 is a source of real concern. Since the first assessments revealed an increased infectivity, it was natural to wonder how the predictions of the various models of the Covid-19 epidemic stand up with this new challenge. On the other hand, since the various media propagate quasi-apocalyptic visions, with predictions often off the top of the head of imaginative reporters, it was imperative to reach some robust conclusions based on reliable models.
The approach we adopted here (and in fact in all our previous studies on Covid-19) was to keep our model as simple as possible. The advantage of such an approach is that in this way one can master the inputs of the model resulting in more robust results. We have thus opted for an as simple as possible version of the SIR model, following Holling's recommendations of a "somewhat
more complex model" [36]. This means that our models tend to be simple, and we enrich them just when necessary, but still shying away from making them too complex. Starting from a simple SIR model, we introduced a seasonally varying virus infectivity in agreement with what has been (and is being) observed in the evolution of the epidemic. The vaccination campaign, which started for most countries at the beginning of this year, was also incorporated into our model, with the possibility of accounting for an incomplete protection from the emerging variant delta. We have also provided for what we have dubbed the recruitment effect, corresponding to local increase of the susceptible population due to travelling. Finally, an extension of the basic, single population, model to one involving more than one age group was introduced along the lines of the approach of [10].
Our main conclusion is that the new variant will probably lead to the appearance of a wave of infections during the summer months. This is what the media refer to as the "fourth wave", the presages of which can be already seen in the evolution of cases in the UK and Portugal. However, and contrary to what is disseminated, the delta-induced wave is not expected to be of distressing consequences (at least for the majority of European countries). The successful vaccination campaign has offered a serious protection to the older, and more vulnerable, population. The delta-related wave is expected to be associated to the younger part of the population and, while, depending on the details of the scenario simulated, the absolute numbers of infected individuals can be appreciable (in particular, if one considers the cumulative effects of vacation-related recruitment and slightly inefficient vaccines), the stress on the health system is expected to be manageable.
Fig. 9. Temporal evolution of the infective fraction of the population, with seasonality, for the two variants, in the case of only one age group, but with a fraction of vaccination deniers. Vaccination starts at t = 2 months with a rate of v/X = 1.8 • 10-2 and an efficiency of 100% (f = 0). The fraction of vaccination deniers is 25 %
The key factor in all this are the vaccinations. It is, for instance, essential that the vaccination rate be maintained during the summer months. Policies aimed at encouraging and facilitating the vaccination of the younger layers of the population will have a most beneficial effect on the mitigation of the epidemic. Which brings us to the last point we wish to touch upon in this work. We have been worried from the outset about the assumption concerning the constant character of the vaccination rate. It is now clear that a certain group of the population are vaccination deniers [37] and would not get vaccinated, despite been confronted with
the devastating consequences of Covid-19. We have thus performed a final simulation where we assumed that 25 % of the population will not get vaccinated.
The results are shown in Fig. 9. Admittedly 25 % is rather high since it lumps together hard-core deniers and hesitant [38]. Be that as it may, the predictions of the simulations give pause for thought. If the vaccinations stop when just 75 % of the population are protected, the fourth wave may develop into a huge one, extending all the way to the end of spring. And, while spread over a larger time interval, the total number of hospitalisations can be comparable to that of the previous wave. Thus, the one recommendation one can formulate based on the results of our simulations is that vaccinations should continue unabated and that no effort should be spared when it comes to convincing those who still hesitate.
References
[1] Landau, E., The Hard Lessons of Modeling the Coronavirus Pandemic, online at https://www.quantamagazine.org/the-hard-lessons-of-modeling-the-coronavirus-pandemic-20210128/
(2021).
[2] Perra, N., Non-Pharmaceutical Interventions during the COVID-19 Pandemic: A Review, Phys. Rep, 2021, vol. 913, pp. 1-52.
[3] Brauner, J. M., Mindermann, S., Sharma, M., Johnston, D., Salvatier, J., Gavenciak, T., Stephenson, A. B., Leech, G., Altman, G., Mikulik, V., Norman, A. J., Monrad, J.T., Besiroglu, T., Ge, H., Hartwick, M. A., Teh, Y. W., Chindelevitch, L., Gal, Ya., and Kulveit, J., Inferring the Effectiveness of Government Interventions against COVID-19, Science, 2021, vol. 371, no. 6531, eabd9338.
[4] European Centre for Disease Prevention and Control, Guidelines for Non-Pharmaceutical Interventions to Reduce the Impact of COVID-19 in the EU/EEA and the UK, online at https://www.ecdc.europa.eu/en/publications-data/covid-19-guidelines-non-pharmaceutical-interventions (24 Sept 2020).
[5] Nakamura, G., Grammaticos, B., and Badoual, M., Confinement Strategies in a Simple SIR Model, Regul. Chaotic Dyn, 2020, vol. 25, no. 6, pp. 509-521.
[6] Fokas, A. S., Cuevas-Maraver, J., and Kevrekidis, P. G., Easing COVID-19 Lockdown Measures While Protecting the Older Restricts the Deaths to the Level of the Full Lockdown, Sci. Rep., 2021, vol. 11, Art. No. 5839.
[7] Li, X., Mukandavire, Ch., Cucunuba, Z.M., Londono, S., Abbas, K., Clapham, H., Jit, M., Johnson, H., Papadopoulos, T., Vynnycky, E., Brisson, M., Carter, E., Clark, A., Villiers, M., Eilert-son, K., Ferrari, M., Gamkrelidze, I., Gaythorpe, K., Grassly, N., Hallett, T., Hinsley, W., Jackson, M., Jean, K., Karachaliou Mmath, A., Klepac, P., Lessler, J., Li, X., Sean, H., Moore, S., Nayagam, Sh., Nguyen, D. M., Razavi, H., Razavi-Shearer, D., Resch, S., Sanderson, C., Sweet, S., Sy, S., Tam, Y., Tanvir, H., Tran, Q.M., Trotter, C., Truelove, Sh., van Zandvoort, K., Verguet, S., Walker, N., Winter, A., Woodruff, K., Ferguson, N., and Garske, T., Estimating the Health Impact of Vaccination against Ten Pathogens in 98 Low-Income and Middle-Income Countries from 2000 to 2030: A Modelling Study, The Lancet, 2021, vol. 397, no. 10272, pp. 398-408.
[8] Mahase, E., Covid-19: Israel Sees New Infections Plummet Following Vaccinations, BMJ, 2021, vol. 372, n338.
[9] COVID-19 Vaccine Breakthrough Infections Reported to CDC — United States, January 1-April 30, 2021, MMWR Morb. Mortal Wkly Rep., 2021, vol. 70, no. 21, pp. 792-793.
[10] Nakamura, G., Grammaticos, B., and Badoual, M., Vaccination Strategies for a Seasonal Epidemic: A Simple SIR Model, Open Commun. Nonlinear Math. Phys., 2021, vol. 1, pp. 20-40.
[11] SARS-CoV-2 Variant Classifications and Definitions, online at https://www.cdc.gov/coronavirus/2019-ncov/variants/variant-info.html (2021).
[12] Campbell, F., Archer, B., Laurenson-Schafer, H., Jinnai, Y., Konings, F., Batra, N., Pavlin, B., Vandemaele, K., Van Kerkhove, M.D., Jombart, Th., Morgan, O., and le Polain de Waroux, O., Increased Transmissibility and Global Spread of SARS-CoV-2 Variants of Concern As at June 2021, Euro Surveill, 2021, vol. 26, no. 24, 2100509.
[13] Davies, N.G., Abbott, S., Barnard, R. C., Jarvis, C.I., Kucharski, A. J., Munday, J.D., Pearson, C. A. B., Russell, T. W., Tully, D. C., Washburne, A. D., Wenseleers, T., Gimma, A., Waites, W., Wong, K.L.M., van Zandvoort, K., Silverman, J.D., CMMID COVID-19 Working Group, COVID-19 Genomics UK (COG-UK) Consortium, Diaz-Ordaz, K., Keogh, R., Eggo, R. M., Funk, S., Jit, M., Atkins, K. E., and Edmunds, W. J., Estimated Transmissibility and Impact of SARS-CoV-2 Lineage B.1.1.7 in England, Science, 2021, vol. 372, no. 6538, eabg3055.
[14] Ito, K., Piantham, C., and Nishiura, H., Predicted Dominance of Variant Delta of SARS-CoV-2 before Tokyo Olympic Games, Japan, July 2021, Euro Surveill, 2021, vol. 26, no. 27, 2100570.
[15] Sheikh, A., McMenamin, J., Taylor, B., and Robertson, C., SARS-CoV-2 Delta VOC in Scotland: Demographics, Risk of Hospital Admission, and Vaccine Effectiveness, The Lancet, 2021, vol. 397, no. 10293, pp. 2461-2462.
[16] Planas, D., Veyer, D., Baidaliuk, A., Staropoli, I., Guivel-Benhassine, F., Rajah, M. M., Planchais, C., Porrot, F., Robillard, N, Puech, J., Prot, M., Gallais, F., Gantner, P., Velay, A., Le Guen, J., Kassis-Chikhani, N., Edriss, D., Belec, L., Seve, A., Courtellemont, L., Pere, H., Hocqueloux, L., Fafi-Kremer, S., Prazuck, T., Mouquet, H., Bruel, T., Simon-Loriere, E., Rey, F. A., and Schwartz, O., Reduced Sensitivity of SARS-CoV-2 Variant Delta to Antibody Neutralization, Nature, 2021, vol. 596, no. 7871, pp. 276-280.
[17] Kermack, W. O. and McKendrick, A. G., Contributions to the Mathematical Theory of Epidemics, Proc. Roy. Soc. Edinburgh Sect. A, 1927, vol. 115, no. 772, pp. 700-721.
[18] Li, R., Li, Y., Zou, Zh., Liu, Y., Li, X., Zhuang, G., Shen, M., and Zhang, L., Projecting the Impact of SARS-CoV-2 Variants on the COVID-19 Epidemic and Social Restoration in the United States: A Mathematical Modelling Study, Preprint, https://doi.org/10.1101/2021.06.24.21259370 (June 2021).
[19] Nakamura, G., Grammaticos, B., and Badoual, M., Recruitment Effects on the Evolution of Epidemics in a Simple SIR Model, Regul. Chaotic Dyn, 2021, vol. 26, no. 3, pp. 305-319.
[20] Aleta, A. and Moreno, Y., Evaluation of the Potential Incidence of COVID-19 and Effectiveness of Containment Measures in Spain: A Data-Driven Approach, BMC Med., 2020, vol. 18, Art. No. 157.
[21] Merow, C. and Urban, M., Seasonality and Uncertainty in Global COVID-19 Growth Rates, Proc. Natl. Acad. Sci. USA, 2020, vol. 117, no. 44, pp. 27456-27464.
[22] Carleton, T., Cornetet, J., Huybers, P., Meng, K., and Proctor, J., Evidence for Ultraviolet Radiation Decreasing COVID-19 Growth Rates: Global Estimates and Seasonal Implications, Proc. Natl. Acad. Sci. USA, 2021, vol. 118, no. 1, e2012370118.
[23] Sajadi, M.M., Habibzadeh, P., Vintzileos, A., Shokouhi, S., Miralles-Wilhelm, F., and Amoroso, A., Temperature, Humidity, and Latitude Analysis to Estimate Potential Spread and Seasonality of Coronavirus Disease 2019 (COVID-19), JAMA Netw. Open, 2020, vol. 3, no. 6, e2011834.
[24] Watanabe, M., Early Detection of Seasonality and Second-Wave Prediction in the COVID-19 Pandemic, Preprint, https://doi.org/10.1101/2020.09.02.20187203 (2020).
[25] Dan, J.M., Mateus, J., Kato, Y., Hastie, K. M., Yu, E.D., Faliti, C.E., Grifoni, A., Ramirez, S.I., Haupt, S., Frazier, A., Nakao, C., Rayaprolu, V., Rawlings, S.A., Peters, B., Krammer, F., Simon, V., Saphire, E. O., Smith, D. M., Weiskopf, D., Sette, A., and Crotty, S., Immunological Memory to SARS-CoV-2 Assessed for Up to 8 Months after Infection, Science, 2021, vol. 371, no. 6529, eabf4063.
[26] Gaebler, C., Wang, Z., Lorenzi, J.C.C., Muecksch, F., Finkin, S., Tokuyama, M., Cho, A., Jankovic, M., Schaefer-Babajew, D., Oliveira, T. Y., Cipolla, M., Viant, C., Barnes, C. O., Bram, Y., Breton, G., Hagglof, T., Mendoza, P., Hurley, A., Turroja, M., Gordon, K., Millard, K.G., Ramos, V., Schmidt, F., Weisblum, Y., Jha, D., Tankelevich, M., Martinez-Delgado, G., Yee, J., Pa-tel, R., Dizon, J., Unson-O'Brien, C., Shimeliovich, I., Robbiani, D.F., Zhao, Z., Gazumyan, A., Schwartz, R. E., Hatziioannou, T., Bjorkman, P. J., Mehandru, S., Bieniasz, P. D., Caskey, M., and Nussenzweig, M.C., Evolution of Antibody Immunity to SARS-CoV-2, Nature, 2021, vol. 591, no. 7851, pp. 639-644.
[27] Abu-Raddad, L. J., Chemaitelly, H., Butt, A. A., and National Study Group for COVID-19 Vaccination, Effectiveness of the BNT162b2 Covid-19 Vaccine against the B.1.1.7 and B.1.351 Variants, N. Engl. J. Med., 2021, vol. 385, no. 2, pp. 187-189.
[28] Shi, W., Tong, C., Zhang, A., Wang, B., Shi, Z., Yao, Y., and Jia, P., An Extended Weight Kernel Density Estimation Model Forecasts COVID-19 Onset Risk and Identifies Spatiotemporal Variations of Lockdown Effects in China, Commun. Biol., 2021, vol. 4, no. 1, Art. No. 126.
[29] Saito, M. M., Imoto, S., Yamaguchi, R., Sato, H., Nakada, H., Kami, M., Miyano, S., and Higuchi, T., Extension and Verification of the SEIR Model on the 2009 Influenza A (H1N1) Pandemic in Japan, Math. Biosci, 2013, vol. 246, no. 1, pp. 47-54.
[30] Zhao, Y., Wood, D. T., Kojouharov, H.V., Kuang, Y., and Dimitrov, D. T., Impact of Population Recruitment on the HIV Epidemics and the Effectiveness of HIV Prevention Interventions, Bull. Math. Biol., 2016, vol. 78, no. 10, pp. 2057-2090.
[31] Grammaticos, B., Willox, R., and Satsuma, J., Revisiting the Human and Nature Dynamics Model, Regul. Chaotic Dyn, 2020, vol. 25, no. 2, pp. 178-198.
[32] Mickens, R. E., Exact Solutions to a Finite-Difference Model of a Nonlinear Reaction-Advection Equation: Implications for Numerical Analysis, Numer. Methods Partial Differential Equations, 1989, vol. 5, no. 4, pp. 313-325.
[33] Grammaticos, B., Ramani, A., Satsuma, J., and Willox, R., Discretising the Painleve Equations a la Hirota-Mickens, J. Math. Phys., 2012, vol. 53, no. 2, 023506, 24 p.
[34] Hethcote, H. W., The Mathematics of Infectious Diseases, SIAM Rev., 2000, vol. 42, no. 4, pp. 599653.
[35] Ramani, A., Grammaticos, B., Satsuma, J., and Willox, R., Discretisation Induced Delays and Their Role in the Dynamics, J. Phys. A, 2008, vol. 41, no. 20, 205204, 12 p.
[36] Holling, C. S., A Journey of Discovery, online at https://www.resalliance.org/files/Buzz_Holling_Me-moir_2006_a_journey_of_discovery_buzz_holling.pdf (2006).
[37] Hotez, P., COVID Vaccines: Time to Confront Anti-Vax Aggression, Nature, 2021, vol. 592, no. 7856, pp. 661-661.
[38] Schwarzinger, M., Watson, V., Arwidson, P., Alla, F., and Luchin, S., COVID-19 Vaccine Hesitancy in a Representative Working-Age Population in France: A Survey Experiment Based on Vaccine Characteristics, Lancet Public Health, 2021, vol. 6, no. 4, E210-E221.
