key: cord-0441377-p77vt9j7
authors: Deeb, Omar El; Jalloul, Maya
title: Efficacy versus abundancy: which vaccination schemes can better prohibit deaths and active infections?
date: 2021-08-31
journal: nan
DOI: nan
sha: fec50bc1cd15a122916b229842deb732dd2b8890
doc_id: 441377
cord_uid: p77vt9j7

In this paper, we introduce a general novel compartmental model accounting for the effects of vaccine efficacy, deployment rates and timing of initiation of deployment. It consists of compartments corresponding to susceptible, vaccinated susceptible, infectious, vaccinated infectious, active, and dead populations with various vaccine efficacies and vaccination deployment rates. Abstract We simulate different scenarios and initial conditions, and we find that the abundance and higher rate of deployment of low efficacy vaccines would lower the cumulative number of deaths in comparison to slower deployment of high efficacy vaccines. However, the latter can lower the number of active cases and achieve faster and higher herd immunity. We also forecast that, at the same daily deployment rate, the earlier introduction of vaccination schemes with lower efficacy would also lower the number of deaths with respect to a delayed introduction of high efficacy vaccines, which can however, still achieve lower numbers of infections and better herd immunity.

well as emergence of new variants.

The spread patterns of infectious diseases have been and are being actively studied and simulated using several mathematical and computational models. A wide variety of techniques were employed ranging from compartmental models, Agent-Based models (ABM), spatio-temporal analysis to data-driven analysis and artificial intelligence [28, 29] . The first compartmental model was the famous Kermack-McKendrick Susceptible-Infectious-Removed (SIR) model which divides the total population under study into three compartments where agents would move from one compartment to another upon infection or removal through recovery or death [30] .

Numerous variations of this model were introduced to account for characteristics and dynamics of different diseases as well as in simulating similar interactions and spreads in the social and behavioral sciences [31, 32, 33] .

The research on COVID-19 spread extensively employed SIR models that were improved to account for other effects like exposure, travel, quarantine, vaccination, public measures and many other alterations to best describe the underlying features of the contagion [34, 35, 36, 37, 38] , as well as general and country specific ABMs, spatio-temporal and data driven studies [39, 40, 41, 42, 43] .

However, with the mass introduction of various globally and nationally recognized vaccines with varying efficacies, and with the varying degrees of availability and logistic capabilities across the globe, an important question arises for decision makers: which vaccine to choose to deploy in a certain country from the available options in that country, given their efficacies, their availability timeline and expected deployment rate of each. This is a primary issue for public health officials aiming at minimizing the number of deaths and/or the number of active infections, that was not thoroughly discussed and analyzed in the literature. This paper fills the gap by studying the trade-off between vaccine efficacy and abundance and then between efficacy and time of availability, and the corresponding expected outcomes for deaths and infections.

The paper is organized as follows: section 2 presents the novel theoretical compartmental model, section 3 puts forward the results and discussions about the efficacy, deployment and timing rates while the conclusion is presented in section 4.

We introduce a novel compartmental model to account for the effect of vaccines, taking into account different vaccination deployment rates (v), efficacies (e) under several scenarios of infection spread rates represented by the reproductive number (R t ), starting with different initial conditions in relation to different numbers of infections, deaths and immune populations in different countries. The SS V II V RD model consists of 6 compartments: The susceptible unvaccinated population S, the susceptible vaccinated population S V who may catch the virus after being vaccinated during the assumed effective immunity period depending on the efficacy of the vaccine, the infectious unvaccinated population I, the infectious vaccinated population I V who already caught the infection after being vaccinated, the recovered population R who are either fully protected by the vaccine during its effective protection time range or who have already recovered from the infection and still possess immunity, and finally the dead D. Each population is normalized with respect to the total population so that N = S + S V + I + I V + R + D = 1. ordinary differential equations given by:

The parameter β is the rate of meeting between susceptible (S) and all infectious people (I + I V ) and it can be determined from the reproductive number of the infectious spread R t and the average rate of recovery of unvaccinated peopleγ R . In this sense, β = γ R R t . The rate of recovery of vaccinated people is given by γ RV and the rate of meeting between vaccinated susceptible people with infectious people is given by β = γ RV R t .

The rate of death of unvaccinated people is γ D and that of vaccinated ones is γ DV . Finally, T V I represents the average time duration of acquired vaccine immunity and T II represents the average duration of acquired post infection immunity. The change in the total population is dN dt = 0, assuming that N remains constant during that period, neglecting other changes due to natural growth, immigration, etc...

The transfer dynamics between different compartments can be summarized according to the following:

• The susceptible S may become infected I upon contact with the infectious (vaccinated or non-vaccinated) are rate β. They may also become recovered R upon receiving a vaccine with efficacy e given at a daily rate v. In addition, recovered R people would become susceptible again after some time of recovery. Mainly, recovered people due to infection would become susceptible again in an infection immunity period T II and recovered people due to vaccination would become susceptible again in a vaccine immunity period T V I .

• The vaccinated susceptible population S V are the susceptibles S who took the vaccine but are still susceptible to infection, and they may become infectious I V upon a meeting rate β with other infectious people.

• The infectious compartment I is populated by unvaccinated susceptibles S bumping into other infectious agents at a rate β, while exit from this compartment is attributed to deaths at rate γ D into compartment D and recoveries at rate γ R into compartment R.

• The vaccinated infectious compartment I V is formed by vaccinated susceptibles S V catching the disease at a rate β and diminished by people dying at rate γ DV into D and people recovering a rate γ RV into R.

• The recovered population R is formed by effectively vaccinated susceptible people coming from S and people surviving the infection at rates γ R and γ RV from infectious and vaccinated infectious populations I and I V . Simultaneously, recovered people would eventually lose their acquired immunity on average periods of T II after infection and T V I after vaccination, thus would exit the recovered compartment R back to the susceptibles S.

• Finally, the dead D increase at death rates γ D and γ DV among infectious and vaccinated infectious populations I and I V .

The vaccination deployment rate v depends on the available supply of the vaccine as well as the logistical capability and the popular demand at a given time [44] . It is taken in this model as the daily vaccination percentage of the susceptible population, with different scenarios representing slow, moderate and fast deployment rates.

The vaccine efficacy varies among different employed vaccines as well as in relation to new emerging variants in addition to the possibility of supplying a single dose of a double dosed vaccine under short supply. The model accounts for a range of scenarios with low, moderate and high efficacy corresponding to the former cases. We also inspect those scenarios under different reproductive rates R t which depend on different values of mitigation measures related to social distancing, protective masks, sanitation, and other factors that alter the rate of infection spread. We assume different scenarios of low, moderate, high and alternating (where R t varies in a periodic pattern between high and low extrema) reproductive rates.

The numerical values of these parameters are taken in relation to available data and research. The efficacy values vary between 0.5 ≤ e ≤ 0.95 depending on the available vaccines [45, 46, 47, 48, 49, 50, 51] . The rate of recovery for vaccinated people γ R = 1 14 in relation of an average of 14 days needed for recovery [52] , while the death rate amounts to around 2% [9] of the infected which leads to γ D = γ R 50 . We also assume that those who are infected after vaccination would need a similar time of recovery, despite the fact that their symptoms would be much reduced [53, 54] , thus γ RV = γ R but their death rate, as studies reveal, would be considerably lower by 70 − 85% [54, 55] , which is modeled through γ DV = γ D 5 . We take T V I = 90 days and T II = 360 days to represent expected periods of acquired immunity after recovery and after vaccination respectively [56] . The deployment rate v is assigned hypothetical values varying between 0.1% of the susceptible population per day for slow Figure 2 : The relative numbers of active cases, infected vaccinated cases, immune population and total cumulative deaths are shown for a low reproductive rate R = 0.7 under initial conditions of: 40.6% vaccinated population 1.8% currently infected and 0.18% dead (upper row) and 3.1% vaccinated population 0.46% currently infected and 0.11% dead (lower row) for nine different vaccination scenarios. The forecast corresponds to vaccine efficacy (e) and daily deployment rate green) , e = 55%, v = 1.5% (in orange), e = 55%, v = 1% (in purple) and e = 55%, v = 0.7% (in yellow).

vaccination rollout and 1.5% for the highest pace of vaccination rollout. The reproduction rate R t is assigned values of 0.7, 1.1 and 1.5 corresponding to low, medium and high reproduction rates, in addition to a fourth scenario where R t alternates sinusoidally between 0.4 and 2 according to the relation R t = 0.8 cos( t 15 ) + 1.2 where t represents the time in days, with a period of around 94 days to simulate the effect of the consecutive waves of the spread of the infection, as observed empirically [57, 58] . We simulate this model under different combinations of efficacy e, deployment rate v and reproductive number R t to analyze the corresponding cumulative numbers of infected, recovered and dead populations.

We simulate the theoretical model introduced in (1) to determine relative numbers of active cases, infected vaccinated cases, immune population and total cumulative deaths under nine different combinations of efficacies (e) of available vaccines and their deployment rates (v) given by: e = 92%, v = 0.5%, e = 92%, v = 0.3%, e = 92%, v = 0.1%, e = 72%, v = 0.7%, e = 72%, v = 0.5%, e = 72%, v = 0.3%, e = 55%, v = 1.5%, e = 55%, v = 1% and e = 55%, v = 0.7% in order to compare the levels of infection, death and immunity between higher efficacy vaccines at lower abundance and lower efficacy vaccines with more abundance or deployment rate, together with middle values between them. We also account for different possible current situations of infection spread that may differ from one country to another by taking two hypothetical initial conditions corresponding to countries which already achieved high levels of vaccination (40.6% vaccinated population 1.8% currently infected and 0.18% dead) and countries with low current levels of vaccination (3.1% vaccinated population 0.46% currently infected and 0.11% dead). In this sense, the results of this forecast are not country specific but are of global significance. We repeat this simulation for different levels of infection spread modeled through low, medium, high and alternating reproductive rates R t defined before. The results are displayed in figures 2,3,4 and 5 respectively. for a country at early spread stages, the two slowest deployments of the low efficacy vaccine lead to the highest infections among the vaccinated population. Immunity will be maximally obtained under the deployment of high efficacy vaccines at high rates, while the least deaths are achieved under the fastest deployment rates of vaccines, like what was shown before for low and medium spread rates, but with a much higher magnitude of deaths in all scenarios due to a higher reproductive rate.

In the more realistic case of an alternating reproductive number varying between 0.4 ≤ R t ≤ 2 and corresponding to consecutive patterns of low and high spread waves, we realize that in a country at a middle spread stage, all vaccination strategies would lead to bringing down the number of active cases with a small peak arising after the return of the next wave, while in a country at early stages of spread, the slow deployment of 

An important aspect of vaccination strategies that we consider in this paper is the question of vaccine efficacy versus time of availability or start of deployment. In this simulation, we forecast the relative number of active cases, infected vaccinated people, the immune population and the cumulative number of deaths under low, medium and high reproductive rates R = 0.7, 1.1 and 1.5 respectively in figures (6, 7 and 8) . Four cases simulating the effect of efficacy and the time of initiation of the vaccination process were considered: a vaccine of low efficacy e = 52% to start deployment immediately (red), a vaccine of medium efficacy e = 72% to start deployment immediately (black), a medium efficacy vaccine with e = 72% to start deployment after 30 days (blue) and a vaccine of high efficacy e = 92% to start deployment after 30 days (brown), all being deployed at an equal rate of the susceptible population per day. As in previous simulations, the upper row corresponds to countries at middle stages of spread with initial conditions of 40.6% vaccinated population 1.8% currently infected and 0.18% dead while the lower row corresponds to countries at early stages with 3.1% vaccinated population 0.46% currently infected and 0.11% dead.

Under the circumstances of low reproductive rate R t = 0.7 depicted in figure (6) , it is clear that the number scenarios, while in early stage countries, the late deployment scenarios would cause a rise and a peak in infections before falling down significantly, whereas the quick deployment of low or medium efficacy vaccines would lower the active cases faster during the first few months. We also notice that in about 100 days, the number of active cases due to a delayed high efficacy vaccine will catch up and fall below the expected active cases under an immediately deployed low efficacy vaccine. The number of infected vaccinated people is the highest for the lowest efficacy vaccine and the lowest for the highest efficacy vaccine in both country categories. Similarly, for both categories, herd immunity is maximally attained using the delayed high efficacy vaccine rather than the immediate low efficacy one which provides the lowest percentile of immune population. However, regarding deaths, the lowest numbers of deaths are attributed to the immediate deployment of medium then low efficacy vaccines, while a delayed deployment will cause more deaths even while using high and medium efficacy vaccines.

Under the scenario of high reproductive number simulated by R t = 1.5, the number of active cases would fall slowly in countries that are in their middle stages of vaccination and spread, while it would rise and peak under all vaccination scenarios in early stage countries. The highest peaks are attributed to delayed vaccinations even though infections would fall rapidly once vaccination starts, and eventually the corresponding number of active cases would fall below that of low efficacy vaccine being deployed immediately. Immediate deployment also helps to flatten the curve on infections, thus reducing the expected peak number of cases. The maximum number of infected vaccinated people would correspond low efficacy vaccine deployed immediately while the lowest corresponds to the high efficacy vaccine deployed late, for both country categories. Similarly, herd immunity is maximally achieved by the high efficacy vaccine despite being introduced late while the lowest level of immunity is caused by the low efficacy vaccine despite early introduction. The immediate deployment of medium efficacy vaccine minimizes the number of deaths, followed by the immediate low efficacy vaccine, while late deployment raises the number of deaths for both country categories, but with higher relative differences among the death outcomes for middle stage countries.

This study takes into account all parameters related to vaccine efficacy and deployment rates under several infectious rates and initial conditions. However, there is a limiting factor related to the maximal proportion of the population who are willing or at least who would eventually take the vaccine. Here, we took this into account indirectly by using a daily deployment rate v which is proportional to the susceptibles S, and not to the total population N . In this sense, in the initial phases of vaccination, the number of people taking the vaccine would be the highest, but as time progresses, the number of susceptibles decreases, hence the daily number of vaccinated people decreases. It is natural to assume that, as when a country reaches a high level of vaccination, less people will be willing to get the vaccine. If vaccination rates were only connected to abundance or logistic infrastructure, they would have been linked to the total population N .

Due to various reasons ranging from religious and political beliefs, into non-scientific and anti-vaxxer fears, there might be a sizable sector of the society who would refuse to get vaccinated [27] . Vaccine hesitancy is not directly simulated in the model, but it is indirectly represented through relating the daily deployment rate to the number of susceptibles hence it decreases as the number of vaccinated people increases. In our scenario, immunity of this portion of the population would still be achieved through infection rather than vaccination.

In this paper we introduced a general novel compartmental model accounting for the vaccinated population, infected vaccinated population, active infections, and deaths with various vaccine efficacies and vaccination deployment rates.

We simulated different scenarios and initial conditions, and we showed that abundance and higher rate of deployment of low efficacy vaccines would lower the cumulative number of deaths in comparison to slower deployment of high efficacy vaccines. However, the high efficacy vaccines can better lower the number of active cases and achieve faster and higher herd immunity.

We also discovered that at the same daily deployment rate, the earlier introduction of vaccines with lower efficacy would also lower the number of deaths with respect to a delayed introduction of high efficacy vaccines, which can, however, lower the number of infections and attain higher levels of herd immunity.

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