Risk assessment of COVID-19 epidemic resurgence in relation to SARS-CoV-2 variants and vaccination passes

Abstract

The introduction of COVID-19 vaccination passes (VPs) by many countries coincided with the Delta variant fast becoming dominant across Europe. A thorough assessment of their impact on epidemic dynamics is still lacking. Here, we propose the VAP-SIRS model that considers possibly lower restrictions for the VP holders than for the rest of the population, imperfect vaccination effectiveness against infection, rates of (re-)vaccination and waning immunity, fraction of never-vaccinated, and the increased transmissibility of the Delta variant. Some predicted epidemic scenarios for realistic parameter values yield new COVID-19 infection waves within two years, and high daily case numbers in the endemic state, even without introducing VPs and granting more freedom to their holders. Still, suitable adaptive policies can avoid unfavorable outcomes. While VP holders could initially be allowed more freedom, the lack of full vaccine effectiveness and increased transmissibility will require accelerated (re-)vaccination, wide-spread immunity surveillance, and/or minimal long-term common restrictions.

Keywords: Epidemiology; Public health.

Fig. 1. The VAP-SIRS model and its predicted scenarios.

a Graphical scheme of the VAP-SIRS model. b, c Predicted scenarios for the reference setup for the Delta variant, with vaccine effectiveness a = 0.79 (corresponding to the effectiveness of the Comirnaty vaccine against infection with the Delta variant), slow (re-)vaccination rate (υ = υ_r = 0.004; typical for many European countries), slow immunity waning ω = 0.002, low fraction of never-vaccinated (d = 0.12; corresponding to the fraction in the United Kingdom) and proportional mixing (see Methods). b Color curves: Timeline of daily incidence per 1 million inhabitants in different infected compartments for the combination of restrictions f = 0.77 and f_v = 0.55. A variable with the asterisk (*) indicates that we consider a daily incidence over the corresponding variable. The dashed lines describe infected who are: non-vaccinated (I^*, yellow), vaccinated who did not gained immunity ($I_1^{*}$), and vaccinated who already lost immunity ($I_2^{*}$). By $I_{\mathrm{\Sigma }}^{*}$ (red, solid line) we mean the sum of all daily infected ($I_D^{*}+I_N^{*}+I_1^{*}+I_2^{*}$). Color bands: 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. Colors correspond to specific subpopulations: non-vaccinated susceptible (S, yellow), vaccinated susceptible who did not gained immunity (S₁, light orange) vaccinated susceptible who already lost immunity (S₂, dark orange), vaccinated immunized (V, green). Moreover, by I_Σ (red) and R_Σ (blue) we denote all infected and all recovered (independently of vaccination result), respectively. c Time evolution of the instantaneous reproduction number ${\mathcal{R}}^{*}$ (y axis) depending on the number of days counted from the start of the vaccination program (x axis), in five different scenarios describing the epidemic evolution: overcritical (+, red, f = 0.77 and f_v = 0.38), subcritical (-, blue, f = 0.92 and f_v = 0.71), initially and eventually overcritical (+ - +, orange, the same restrictions as in b: f = 0.77 and f_v = 0.55), eventually overcritical (-+, pink, f = 0.92 and f_v = 0.38), and eventually subcritical (+-, cyan, with f = 0.77 and f_v = 0.71). As controls, two additional scenarios of the epidemic evolution are presented, corresponding to no implementation of VPs and no changes in behavior due to vaccination: subcritical (another example of - scenario, green) with f = f_v = 0.92 and eventually subcritical (another example of +- scenario, yellow) with f = f_v = 0.77, both plotted with dot-dashed line.

Fig. 2. Possible COVID-19 epidemic dynamics for different parameter setups for the Delta variant.

The relevant f − f_v parameter space, where f_v ≤ f, can be divided into five regions (delimited by black borders), each associated with a different behavior of the epidemics. On the diagonal (white dashed line), f = f_v, i.e., the restrictions for VP holders and for the rest of the population are the same - corresponding to the situation when VPs are not introduced at all. Lower triangles show the time until the last critical threshold: different color scales correspond to the time until the switch either from a subcritical to an overcritical epidemic (time until overcriticality, violet-green scale), or from an overcritical to a subcritical epidemic (time until subcriticality, yellow-pink scale). Upper triangles show the asymptotic ${\mathcal{R}}^{*}$, as a function of the values of f and f_v (blue-red scale, with blue associated with ${\mathcal{R}}^{*}<1$ and red associated with ${\mathcal{R}}^{*}>1$). a Reference setup, with a = 0.79 (corresponding to the effectiveness of the Comirnaty vaccine on the Delta variant), υ = υ_r = 0.004, ω = 0.002, d = 0.12 (fraction of never-vaccinated in the United Kingdom) and proportional mixing. The choices of (f, f_v) corresponding to the five scenarios exemplified in Fig. 1c are denoted by points of the same color. b Setup with a decreased vaccine effectiveness: a = 0.6 (corresponding to the effectiveness of the Vaxzevria vaccine on the Delta variant). c Setup with an increased vaccination rate: υ = υ_r = 0.008. d Setup with preferential (instead of proportional) mixing. e Setup with an increased fraction of people who will not get vaccinated: d = 0.3 (fraction of never-vaccinated in France). f Setup with an increased waning rate: ω = 1/200.

Fig. 3. Daily COVID-19 infection cases in the endemic state for different parameter setups and the Delta variant.

Lower triangles show the daily infection numbers in the vaccinated, and upper triangles in the unvaccinated population in the endemic state of the epidemics, for the relevant f − f_v parameter space, where f_v ≤ f. The color scale spans from no more than 10 (yellow) up to 1000 and more daily cases per million people (dark violet). Parameter setups as well as the black borders that delimit the five regions are defined as in Fig. 2: a Reference setup, with a = 0.79 (corresponding to the effectiveness of the Comirnaty vaccine on the Delta variant), υ = υ_r = 0.004, ω = 0.002, d = 0.12 (fraction of never-vaccinated in the United Kingdom) and proportional mixing. b Setup with a decreased vaccine effectiveness: a = 0.6 (corresponding to the effectiveness of the Vaxzevria vaccine on the Delta variant). c Setup with an increased vaccination rate: υ = υ_r = 0.008. d Setup with preferential (instead of proportional) mixing. e Setup with an increased fraction of people who will not get vaccinated: d = 0.3 (fraction of never-vaccinated in France). f Setup with an increased waning rate: ω = 1/200.

Fig. 4. Daily COVID-19 hospitalized cases in the endemic state for different parameter setups and the Delta variant.

Lower triangles show the daily hospitalized numbers in the vaccinated population, and upper triangles in the unvaccinated population, in the endemic state of the epidemic, for the relevant f − f_v parameter space, where f_v ≤ f. The color scale spans from no more than 1 (yellow) up to 100 and more daily hospitalized cases per million people (navy blue). Parameter setups as well as the black borders that delimit the five regions are defined as in Fig. 2: a Reference setup, with a = 0.79 (corresponding to the effectiveness of the Comirnaty vaccine on the Delta variant), υ = υ_r = 0.004, ω = 0.002, d = 0.12 (fraction of never-vaccinated in the United Kingdom) and proportional mixing. b Setup with a decreased vaccine effectiveness: a = 0.6 (corresponding to the effectiveness of the Vaxzevria vaccine on the Delta variant). c Setup with an increased vaccination rate: υ = υ_r = 0.008. d Setup with preferential (instead of proportional) mixing. e Setup with an increased fraction of people who will not get vaccinated: d = 0.3 (fraction of never-vaccinated in France). f Setup with an increased waning rate: ω = 1/200.

Fig. 5. The estimated benefit of vaccination passes (VPs).

For each parameter setting (reference setup and single or double parameter value changes to it; y axis), the benefit of VPs was evaluated (x axis) for two SARS-CoV-2 strains: Alpha (green) and Delta (red).