key: cord-0916864-2zb8rd8p
authors: Krueger, T.; Gogolewski, K.; Bodych, M.; Gambin, A.; Giordano, G.; Cuschieri, S.; Czypionka, T.; Perc, M.; Petelos, E.; Rosinska, M.; Szczurek, E.
title: Risk of COVID-19 epidemic resurgence with the introduction of vaccination passes
date: 2021-05-14
journal: nan
DOI: 10.1101/2021.05.07.21256847
sha: a21128f5e9bddd3dc678c24f0b249ec25a14d6e7
doc_id: 916864
cord_uid: 2zb8rd8p

Many countries hit by the COVID-19 epidemic consider the introduction of vaccination passes. So far, no thorough impact assessment of vaccination passes and of lower restrictions for their holders has been conducted. Here, we propose the VAP-SIRS model that accounts for susceptible, infected, and recovered subpopulations, also within the group of vaccinated pass holders. The model accounts for imperfect vaccination effectiveness, revaccinations and waning immunity. Different restrictions for pass holders and the rest of the population result in different scenarios of the epidemic evolution, some of which yield unfavourable COVID-19 dynamics and new waves. We identify critical variables that should be considered by policymakers and show how unfavourable outcomes can be avoided using adaptive policies. In particular, while pass holders could initially be allowed large freedoms, the gradual loss of immunity will require either increased restrictions for pass holders, or accelerated revaccination. In the long-term, common restrictions for both the pass holders and the rest of the population will have to be kept to avoid epidemic resurgence. Such minimum required restrictions depend on vaccination effectiveness, revaccination rate, waning rate and fraction of never-vaccinated population, and, for realistic combinations of these parameters, range between 29% and 69% reduction of contacts.

program starts, followed by a large wave later (Fig. 1b) . This behavior is explained by the 106 population structure and different restrictions (Fig. 1c) . In this scenario, the first wave is driven 107 by the unvaccinated susceptibles (S N ) and suppressed by ongoing vaccination, as expected. 108 Interestingly, the second, larger wave is driven by the S V group. The S V group is composed of 109 the number of individuals for whom the vaccine was ineffective (S 1 ) and those vaccinated who 110 lose their immunity and are not yet revaccinated (S 2 ). In the following we investigate how the 111 dynamics change for different restrictions settings. Figure 1 :

The VAP-SIRS model shows the possible rebound of infections after a large population is vaccinated and obtains a VP, but then loses immunity. a. Graphical scheme of the VAP-SIRS model. b. The timeline of daily incidence per 1 million inhabitants for an effective vaccine (a = 0.9), slow (re-)vaccination rate (υ = υ r = 0.004; typical for many European countries), proportional mixing (see Methods) and low fraction of never-vaccinated (d = 0.1). Here, a variable with the asterisk ( * ) indicates that we consider a daily incidence over the corresponding variable, thus I * stands for I * D + I * N , and by I * Σ we mean the sum of all daily infected (I * + I * 1 + I * 2 ). c. The Muller plot of the population structure (the width of the color band in the y axis) as a function of time (x axis) for the same parameter settings as in b. Here, by I Σ and R Σ we denote I + I V and R + R V , respectively. 5 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 14, 2021. ; https://doi.org/10.1101/2021.05.07.21256847 doi: medRxiv preprint Stability analysis identifies potential scenarios for the COVID-19 epidemic depending on the restrictions imposed on VP holders and the rest of the 114 population. 115 To assess the epidemic evolution in different scenarios, we analysed stability by linearising the CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 14, 2021. ; https://doi.org/10.1101/2021.05.07.21256847 doi: medRxiv preprint vaccinated population (Fig. 3e ) and higher waning rate (Fig. 3f) , while it shrinks with higher vaccination rate (Fig. 3c) . The subcritical region -(with R * always smaller than 1) lies in 153 the opposite corner of the f − f v space, for larger values, and for a fixed fraction of never-154 vaccinated d tends to decrease for setups where the overcritical region increases. Inside each of 155 the remaining three regions (associated with the +-+, -+, +-scenarios in Figure 2 ), the specific 156 parameter settings differ by the time to the critical threshold of interest for that region (the last 157 observed switch between subcritical and overcritical epidemic, which for the +-+ region, for 158 example, is the second critical threshold; see Methods for the computation of the times to critical 159 thresholds). For the reference setup (Fig. 3a) , the critical threshold is reached after a minimum 160 ∼ 8 months. Decreasing vaccine effectiveness (Fig. 3b) , as well as increasing the waning rate 161 (Fig. 3f) , enlarges the +-+ region' and leads to overcriticality sooner, after ∼ 3 and ∼ 4 months 162 respectively, for low f v values. Increasing vaccination rate (Fig. 3c) shrinks the +-+ region.

With preferential mixing (Fig. 3d) , the +-+ region becomes larger and overcriticality is reached 164 even sooner. Increasing the number of never-vaccinated people (Fig. 3e) shrinks the +-+ region 165 and delays the onset of overcriticality. For all considered parameter setups, except for the one 166 with high (re-)vaccination rate, for all except the +-and theregions, large asymptotic R * can 167 be expected, which corresponds to short doubling times (Fig. 3) . 168 These results indicate that VP holders can be granted large freedom, as long as sufficient 169 restrictions are enforced for the rest of the population, to avoid an initially overcritical situation. 170 However, to prevent the epidemic from becoming overcritical after an initial decline in case 171 numbers, restrictions on VP holders need to be timely increased and adapted so that eventually 172 everyone faces the same restrictions. Safe restrictions correspond to the parameters in the sub-173 critical region, but these are relatively high and could be unacceptable for the population. The

-+ and +-+ regions can seem attractive from the aspect of large freedom for the VP holders.

Both these regions, however, eventually result in epidemic resurgence and should be avoided.

Moving to the +-region with the right timing is a recommendable strategy. 

. CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 14, 2021. ; https://doi.org/10.1101/2021.05.07.21256847 doi: medRxiv preprint

We compute the minimum common restriction level f min for the whole population that would 180 be required to avoid an overcritical epidemic in the long-term (Methods):

where V as is the asymptotic fraction of the immunized in the population

The resulting values differ depending on the setups of vaccine effectiveness a, revaccination 183 rate υ r , the fraction of never-vaccinated population d and immunity waning rate ω (Tab. 1).

Even for the most optimistic setup (high a = 0.9, high υ r = 0.008, low d = 0.1, low ω = The main driver of this phenomenon is the potential lack of immunity of VP holders. With a 196 VP, people enjoy low restrictions while actually being susceptible and potentially contagious 197 because the vaccine was ineffective or the immunity has waned.

VAP-SIRS deliberately keeps several aspects simple (see Methods for model limitations).

10 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Table 1 : Asymptotic level of immunization V as and minimum common restrictions f min for different parameter setups, for parameters: vaccine effectiveness a, revaccination rate υ r , fraction of never-vaccinated d, and waning immunity rate ω. The first row concerns the reference setup; rows below are setups with the same parameters as in the reference setup, but with either one parameter changed (in bold; rows 2-5; same as in Figure 3 , apart from preferential mixing, as it is not relevant for common restrictions) or two parameters changed (in bold; rows 6-11).

The advantage of our analysis is the relevance for long-term dynamics, and the focus on avoid- restriction level, which could also be achieved through temporary VPs. Our model implies that 210 such common restrictions need to be higher than those initially granted to VP holders, and need 211 to be introduced in time to avoid another wave.

As expected, the model shows that there is a larger selection of admissible restrictions' 213 setups under high vaccine effectiveness, slowly waning immunity, proportional social mixing, 214 low share of never-vaccinated and higher vaccination rate. At least the latter two parameters 215 are amenable to policy action. Thus, efforts to increase (re-) vaccination speed and encour-216 11 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 14, 2021. ; https://doi.org/10.1101/2021.05.07.21256847 doi: medRxiv preprint age people to get vaccinated also extend the margin of lowering restrictions for VP holders as well as eventual common restrictions. Social mixing patterns, such as preferential mixing, can 218 accelerate the infection resurgence in time and, although difficult to change by policymaking, 219 should be monitored. Finally, it is noteworthy that VP holders are less likely to be tested, as they 220 are assumed to be protected and they may exhibit milder symptoms. Therefore, their potential 221 infection is more likely to remain undetected, resulting in an effect similar to that of lowering 222 restrictions. To prevent undesirable outcomes, the testing criteria should not exclude the VP 223 holders. In addition, the VP holders should be widely and regularly tested for antibody level, vaccinated from the V group move to the S 2 group of susceptibles with immunity waning rate 18 ω. The individuals from the S 1 group move to S 2 with the same rate ω. The S 2 group is the 19 group of vaccinated, but no longer immune, and thus, susceptible individuals. In contrast to 20 S 1 , we consider that the S 2 group is subject to revaccination. Consequently, a fraction of size 21 a of the population from S 2 populates V with rate aυ r and a fraction of size (1 − a) populates 22 S 1 with rate (1 − a)υ r . Across the manuscript, we assume υ r = υ, but the model is general 23 and different values can be considered. Individuals from S 1 move to S 2 with rate ω to ensure 24 that the ineffectively vaccinated are revaccinated with the same speed as the ones for which the 25 vaccine was effective.

Some of the susceptibles in S 1 (or, similarly, S 2 ) may not get revaccinated fast enough and 27 may become infected and populate I 1 (or, I 2 ). Then, as in the classical SIRS model, the I 1 (or 28 1 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 14, 2021. ; https://doi.org/10.1101/2021.05.07.21256847 doi: medRxiv preprint I 2 ) population recovers and populates group R V with rate γ. We consider that the recovered in 29 R V may also lose the immunity, and become susceptible again and move to S 2 with rate κ. We 30 fix κ to 0.002, corresponding to average 500 days duration of natural immunity. There remains 31 uncertainty regarding the waning time for natural immunity, but early evidence indicates it lasts 32 at least 180 days [2, 3, 4]. Hence, we assume an optimistic scenario of natural immunity lasting 33 similarly long as the immunity gained via vaccination. Before the recovered in the R V lose 34 immunity, they might be revaccinated, and, thus, populate the V group with rate υ r .

The remaining susceptible subgroups (the S N and S D ) may undergo the same classical 36 dynamics, i.e., become infected, recover, and either become susceptible again or, in case of 37 the recovered in the R N subgroup, become vaccinated with rate υ.

Note that for the recovered in the R V or R N groups we assume that vaccination effectiveness 39 is 1, which is substantiated on the basis of the fact that vaccination combined with a previous 40 infection should confer a much stronger protection than only vaccination of a susceptible individual.

The following parameters are used to describe population dynamics in the model: CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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where also the following relations hold 44 S V = S 1 + S 2 ,

with the constraint S, S V , I, I V , R, R V ≥ 0. Finally, to consider the subpopulation dynamics in 45 terms of fractions of the entire subpopulation, we set

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and denote d to be the fraction of the never-vaccinated population

Modeling restrictions 47 We assume that the VP holders consist of the following subpopulations of vaccinated at least The parameter f satisfies f ≥ f v and corresponds to restriction of contacts within the rest of 52 the population, as well as between the VP holders and the rest of the population.

The restriction level f v for the VP holders is introduced in the model as a modulator of the 54 transmission rate β v . Specifically, we assume that to contact other VP holders, since they have lower restrictions [5] . This preferential bias is CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 14, 2021. ; https://doi.org/10.1101/2021.05.07.21256847 doi: medRxiv preprint following rules:

where S + I + R is the non-immune population. 

Thus, the I * (t) is computed as

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We proceed similarly to obtain daily incidence numbers I * 1 , I * 2 and for the sum of all infected, 85 and again to make it interpretable in the figures we re-scale it by 1M. 

For convenience, where it is not needed, we drop the time argument.

Taking an adiabatic approach we linearize the infection dynamics for small I, I V and R under the assumption of slowly varying S, S V and V . In that case, the infection dynamics decouples from the vaccination dynamics and the Jacobian submatrix J sub for the equations for I and I V is given by:

Given the Jacobian submatrix, we can approximate the dynamics in a small neighborhood of

The instantaneous reproduction number R * and the instantaneous doubling time D

Since the largest and the second largest eigenvalues λ max and λ 2 of J sub are both real, the 

The largest eigenvalue of J sub is given by

whereby it is convenient to express λ max as a function of R 1 = β γ and R 2 = βv γ . We then obtain

Given the population fractions S(t) and S V (t) at a given time instant t, the linearized then be understood as a mean field limit of the microdynamics described by such a branching 109 process. Moreover, the spectral norm

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The above discrete branching process can be extended to a continuous time branching process by assuming a probability distribution on the generation time, denoted ϕ(γ). The growth of the continuous time branching process const · e αt is characterized by its Malthusian growth parameter, denoted α. The relation between the instantaneous reproduction number R * , the distribution ϕ (τ ) and the Malthusian parameter α for such a branching process is given by

where L ϕ (α) is the Laplace transform 

By Equation 5, the Malthusian parameter α for our dynamics is given by the largest eigenvalue 119 λ max . Hence we obtain the relation between the instantaneous reproduction R * and the λ max as 120 λ max = γ (R * − 1) . Note that since both S and S V are functions of time, so are λ max and R * .

It is noteworthy that in the above equations, all R 1 , R 2 , R 1 S and R 2 S V , and R * should be 122 seen as reproduction numbers, but of a different nature [9]. R 1 and R 2 are reproduction numbers 123 taking into account the restrictions f and f v , respectively. The R 1 S and R 2 S V are also group 124 specific, but in addition incorporate the respective group sizes. Finally, R * combines all these 125 factors together. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The analysis of the linearized dynamics around I = I V = 0 allows us to determine transitions between subcritical and overcritical epidemics. Such transitions occur at the time instants t at which λ max (t) = 0, or, equivalently, at R * (t) = 1. We thus find that for given values of S(t) and S V (t) the critical times t for transitions between subcritical and overcritical epidemics are the roots of the equation

The obtained critical threshold times are plotted in the lower triangles of the panels in Figure 3 131 in the main text. In the case of proportional mixing the above equation is equivalent to:

Asymptotic structure of the population 133 The asymptotic structure of the population in terms of the sizes of the subpopulations V, S V 134 and S D can be easily obtained by setting I = I V = R = R V = 0 and computing the stable 135 stationary solution for V as , S as and S as V of our ODE system (1):

where η = a 1 + ω/υ r can be seen as the actual immunization rate in the population, and is expressed as a function 137 of vaccine effectiveness a and the ratio of the immunity waning rate ω and the revaccination 138 rate υ r . The obtained values correspond to the structure in the limit t → ∞ and represent the 139 structure to which the population converges in the long term. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 14, 2021. ; the values we considered. Asymptotic endemic states of the ODE system (1) could easily be computed, but are not discussed here due to space constraints. Finally, another limitation of 165 our analysis is that not all parameter values are exactly known, such as the post-vaccination 166 or natural immunity waning time. We, however, fix optimistic values for such parameters, and 167 show that unfavorable infection dynamics can still be obtained even under optimistic assumptions. 

Modeling Infectious Diseases in Humans and Animals

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Van den Driessche, P. Reproduction numbers of infectious disease models

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SC acknowledges support by University of Malta. TC has received funding from the European

The times of transitions between subcritical and overcritical epidemics End notes