key: cord-320360-2qpcot6p
authors: Gumel, A. B.; Iboi, E. A.; Ngonghala, C. N.
title: Will an imperfect vaccine curtail the COVID-19 pandemic in the U.S.?
date: 2020-05-14
journal: nan
DOI: 10.1101/2020.05.10.20097428
sha: 
doc_id: 320360
cord_uid: 2qpcot6p

The novel coronavirus (COVID-19) that emerged from Wuhan city of China in late December 2019 continue to pose devastating public health and economic challenges across the world. Although the community-wide implementation of basic non-pharmaceutical intervention measures, such as social-distancing, quarantine of suspected COVID-19 cases, isolation of confirmed cases, use of face masks in public, and contact-tracing, have been quite effective in curtailing and mitigating the burden of the pandemic, it is universally believed that the use of an anti-COVID-19 vaccine is necessary to build the community herd immunity needed to effectively control and eliminate the pandemic. This study is based on the design and use of a mathematical model for assessing the population-level impact of a hypothetical imperfect anti-COVID-19 vaccine on the control of COVID-19. An analytical expression for the minimum number of unvaccinated susceptible individuals needed to be vaccinated to achieve vaccine-induced community herd immunity is derived. The epidemiological consequence of the herd immunity threshold is that the disease can be effectively controlled or eliminated if the minimum herd immunity threshold is achieved in the community. Simulations of the model, using baseline parameter values obtained from fitting the model with mortality data relevant to COVID-19 dynamics in the US states of New York and Florida, as well as for the entire US, show that, for an anti-COVID-19 vaccine with an assumed protective efficacy of 80%, the minimum herd immunity threshold for the entire US, state of New York and state of Florida are, respectively, 90%, 84% and 85%. Furthermore, it was shown that, while a significantly large increase in vaccination rate (from baseline) is necessarily needed to eliminate COVID-19 from the entire US, the pandemic can be eliminated from the states of New York and Florida if the vaccination rate is marginally increased (by as low as 10%) from its baseline value. The prospect of COVID-19 elimination in the US or in the two states of New York and Florida is greatly enhanced if the vaccination program is combined with a public mask use program or an effective social-distancing measure. Such combination of strategies significantly reduces the vaccine-induced herd immunity threshold. Finally, it is shown that the vaccination program is more likely to lead to COVID-19 elimination in the state of Florida, followed by the state of New York and then the entire US.

In December 2019, the Wuhan Municipal Health Commission, China, reported a cluster of a pneumonia-like illness, caused by a coronavirus (SARS-CoV) in Wuhan, Hubei Province of China [1] [2] [3] . The disease rapidly spread to many countries, resulting in the World Health Organization declaring the novel coronavirus (COVID-19) as a pandemic on March 11, 2020 [1, 2, 4] . As of May 8, 2020, COVID-19 has caused over 4 million confirmed cases 19 vaccine on the transmission dynamics of the COVID-19 pandemic. The model will be parametrized using available COVID-19 mortality data for the US, the state of New York, and the state of Florida to estimate important parameters related to vaccination and other control intervention (notably the parameter related to the reduction in community contacts).

We develop a basic model for assessing the population-level impact of a potential hypothetical anti-COVID-19 vaccine. The model is developed by splitting the total human population at time t, denoted by N (t), into the mutually-exclusive compartments of unvaccinated susceptible (S u (t)), vaccinated susceptible (S v (t)), exposed (E(t)), symptomatically-infectious (I s (t), asymptomatically-infectious (I a (t)), hospitalized (I h (t)) and recovered (R(t)) individuals. Thus,

It should be emphasized that the epidemiological compartment I a also contain individuals who may show mild symptoms of the disease. Furthermore, the compartment I h for hospitalization also includes those with clinical symptoms of COVID-19 who are self-isolating at home. The model is given by the following deterministic system of nonlinear differential equations (where a dot represents differentiation with respect to time t): A flow diagram of the model is depicted in Figure 1 . The state variables and parameters of the model are described in Tables 1 and 2 , respectively. In the model (2.1), the parameter β s (β a ) represents the effective contact rate (i.e., contacts capable of leading to COVID-19 transmission) for symptomatically-infectious (asymptomaticallyinfectious) individuals, 0 < c M ≤ 1 is the proportion of members of the public who wear face-masks (correctly and consistently) in public and 0 < M ≤ 1 is the efficacy of the face-masks. This formulation (i.e., β s = β a ) allows us to account for the possible heterogeneity in the contact rates of infectious with (I s ) or without (I a ) clinical symptoms of COVID-19. Unvaccinated susceptible individuals are vaccinated at a per capita rate ξ v . It is assumed that the hypothetical anti-COVID-19 vaccine is imperfect (i.e., it allows for breakthrough infected in vaccinated susceptible individuals who became infected with COVID- 19) , with a protective efficacy 0 < ε v ≤ 1 against COVID-19. The parameter σ represents the progression rate of exposed individuals (i.e., 1/σ is the intrinsic incubation period of . A proportion, 0 < r ≤ 1, of exposed individuals show no clinical symptoms of COVID-19 (and move to the class I a ) at the end of the incubation period. The remaining proportion, 1 − r, show clinical symptoms and move to the I s class. The parameter γ s (γ a )(γ h ) represents the recovery rate for individuals in the I s (I a )(I h ) class. Similarly, φ s is the hospitalization (or self-isolation) rate of individuals with clinical symptoms of COVID-19. Finally, the parameter δ s (δ h ) represents the COVID-induced mortality rate for individuals in the I s (I h ) class. It should be mentioned that, since not much data is available on the expected properties of a future anti-COVID-19 vaccine, we do not assume that the vaccine offers any therapeutic benefits. Further, we do not assume that the vaccine wanes during the one-year time period we would choose for the numerical simulations of our model. The model (2.1) is an extension of numerous deterministic models for the transmission dynamics of COVID-19 pandemic, such as those in [4, 11] , by explicitly incorporating an imperfect anti-COVID-19 vaccine. To the best of our knowledge, this is the first deterministic model for COVID-19 transmission that explicitly incorporates an imperfect vaccine. This article is a US Government work. It is not subject to copyright under 17 USC 105 and is also made available (which was not certified by peer review) 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, 2020. . Vaccine efficacy against acquisition of infection σ Rate of progression from exposed class to infectious class (1/σ is the incubation period) 1 − r Fraction of exposed individuals who show clinical symptoms at the end of the incubation period γ s (γ a )(γ h ) Recovery rate for individuals in the I s (I a )(I h ) class δ s (δ h ) Disease-induced mortality rate for infectious (hospitalized) individuals φ s Hospitalization rate for infectious individuals 3 Results

The model (2.1) has a continuum of disease-free equilibria (DFE), given by

where S u (0) and S v (0) are the initial total sizes of the populations unvaccinated and vaccinated susceptible individuals, respectively (so that, N (0) = S u (0) + S v (0)). The next generation operator method [17, 18] can be used to analyse the asymptotic stability property of the DFE. In particular, using the notation in [17] , it follows that the associated next generation matrices, F and V , for the new infection terms and the transition terms, are given, respectively, by

and,

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The copyright holder for this preprint this version posted May 14, 2020. . https://doi.org/10.1101/2020.05. 10.20097428 doi: medRxiv preprint It is convenient to define the quantity R c by:

where ρ is the spectral radius,

The quantity R c is the control reproduction number of the model (2.1). It measures the average number of new COVID-19 cases generated by a typical infectious individual introduced into a population where a certain fraction is protected, e.g., vaccinated (with the hypothetical vaccine) against COVID-19. It is the sum of the constituent reproduction numbers associated with the number of new cases generated by symptomatically-infectious humans (R Is ) and asymptomatically-infectious (R Ia ) individuals. The reproduction number R c will now be interpreted epidemiologically below.

As stated above, the reproduction number R c is the sum of the two constituent reproduction numbers, R Is and R Ia . The constituent reproduction number R Is is given by the product of the infection rate of unvaccinated and vaccinated susceptible individuals by symptomatically-infectious humans, near the DFE

, the proportion of exposed individuals that survived the incubation period and moved to the symptomaticallyinfectious class (1−r)σ σ = 1 − r and the average duration in the asymptomatically-infectious class , the proportion that survived the exposed class and moved to the asymptomatically-infectious class rσ σ = r and the average duration in the I a class 1 γa . The sum of R Is and R Ia gives R v . The result below follows from Theorem 2 of [17] . The epidemiological implication of Theorem 3.1 is that a small influx of COVID-19 cases will not generate a COVID-19 outbreak if the control reproduction number (R c ) is less than unity.

The basic reproduction number of the model (2.1) (denoted by R 0 ), obtained by setting the control-related parameters and state variables (i.e., c M , ε M , ε v and S * v ) to zero in (3.1), is given by

The quantity R 0 represents the average number of secondary infections generated by a typical infected individual introduced into a completely susceptible population during the duration of the infectiousness of the individual.

Some of the parameters for Model (2.1) were extracted (or estimated based on COVID-19 information) from the literature (Table 3 (a)), while the rest (see Table 3 (b)) were estimated using mortality data for the entire US, the state of New York, and Florida [19] [20] [21] [22] . A nonlinear least squares method was used to determine the set of parameters that minimizes the sum of the squares of the differences between the cumulative death predictions of the model 

and cumulative COVID-induced mortality) with increasing effectiveness of social-distancing, as expected. For instance, while Figure 3 shows that up to 120,000 cumulative mortality can be recorded in the US if social-distancing measures were not implemented, the cumulative mortality dramatically decreases to about 64,000 (representing about 47% reduction in cumulative mortality) if strict social-distancing measures were implemented (Figure 3 (b)). Similarly, 80% and 70% reductions in cumulative mortality are, respectively, obtained for the states of New York (Figures 3 (c) and (d)) and Florida (Figures 3 (e) and (f)). These results are consistent with those reported by Ngonghala et al. [4] , which was based on using a novel mathematical model that incorporates the main nonpharmaceutical interventions against COVID-19 (i.e., the model in [4] does not incorporate an anti-COVID-19 vaccine). This article is a US Government work. It is not subject to copyright under 17 USC 105 and is also made available (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 

For vaccine-preventable diseases, not all susceptible individuals can be immunized for various reasons, such as they are too young to be vaccinated (vaccinating infants or young children can, sometimes, harm the infants/children), they have co-morbidities and weakened immune system (and vaccinating may make their prognosis worse), they are of advanced age or they opt out for cultural, traditional or religious reasons. The question then is what is the minimum proportion of those we can vaccinate that we need to vaccinate so that those that we cannot vaccinate can be protected from developing severe disease or dying of the disease. The notion of herd immunity in the transmission dynamics is associated with the indirect protection, against acquisition of a communicable disease, members of the community receive when a large percentage of the population has become immune to the infectious disease due to natural recovery from prior infection or vaccination [30, 31] . The consequence of herd immunity is that individuals who are not immune (e.g., those who cannot be vaccinated or those who have not been infected yet) receive some protection against the acquisition of the disease. The safest and fastest way to achieve herd immunity is obviously vaccination. It should, however, be mentioned that Sweden employed the other mechanism for achieving herd immunity in the context of COVID-19 dynamics in Sweden [32] . In other words, the Swedish public health agencies aimed to build herd immunity by not implementing the basic community transmission reduction strategies (e.g., social-distancing, community lockdowns, use of face masks in public, contact tracing etc.) implemented in almost every country or community that is hard-hit with the COVID-19 pandemic, opting, instead, to allow individuals to acquire infection and, hopefully, recover from it. Although some level of this natural herd immunity has been achieved in Sweden, the COVID-induced mortality recorded in Sweden far exceed those recorded for neighbouring Scandinavian countries that implemented the aforementioned non-pharmaceutical interventions [32] .

In this section, a theoretical condition for achieving community-wide vaccine-induced herd immunity is derived. Let f v be the fraction of unvaccinated susceptible individuals that have been vaccinated at the disease-free steady-state. That is,

Further, let

It follows from (3.1), (3.4) and (3.5) that the control reproduction number (R c ) can then be re-written, in terms of f v and g v , as follows:

which can be simplified as

Setting R c = 1 in (3.7), and solving for f v , gives the following vaccine-induced herd immunity threshold for the model (2.1):

It follows from (3.7) and (3.8) that R c < (>)1 whenever f v > (<)f c v . Hence, we achieved the following results:

Theorem 3.2. Consider the model (2.1). The use of an imperfect anti-COVID-19 vaccine will have the following

The result of Theorem 3.2 is numerically-illustrated in Figure 4 , showing an increase in the cumulative COVIDinduced mortality when f v < f c v and a decrease when f v > f c v . In order to compute the community herd immunity thresholds for the US, New York state and Florida state, we compute the respective values of R 0 , using (3.8) with the baseline values of the community contact rates parameters, β a and β s , reduced by 30% (to account for the effect of the implementation of effective social-distancing measures [4] ). That is, we set β a = 0.8400, β s = 0.4971 for US, β a = 0.5648, β s = 0.3871 for New York state and β a = 0.7675, β s = 0.4621 for the state of Florida in Equation (3.8) to obtain the values of the basic reproduction number (R 0 ) for the US, New York state and the state of Florida. This gives R 0 = 3.58, 3.04 and 3.12, respectively. Substituting these values of R 0 (for each of the three populations) in Equation (3.7), and noting that the vaccine efficacy is assumed to be 80% (i.e., ε v = 0.8), shows that the respective vaccine-derived community herd immunity thresholds for the entire US, the state of New York and the state of Florida are 90%, 84% and 85%, respectively (as tabulated in Table 4 ). The epidemiological implication of these analyses is that the pandemic can be effectively controlled and/or eliminated in the US and the states of New York and Florida if at least 90%, 84% and 85% of the respective unvaccinated susceptible populations are vaccinated (using the hypothetical imperfect vaccine with the assumed protective efficacy of 80%). It should be mentioned that these vaccine-derived community herd immunity percentages significantly decrease if the vaccination program is complemented with a public face mask use strategy. In particular, Table 4 shows that the herd immunity threshold for the US reduced to about 78% if half the US population wears mask in public . Face mask usage with compliance of 30% to 50% decreases the herd immunity thresholds for the states of New York and Florida to between 70% to 78%. These are reasonably attainable in the two states. Thus, these simulations show that combining a vaccination program (using the hypothetical anti-COVID-19 vaccine with assumed efficacy of 80%) with a public mask use strategy significantly enhances the prospect of COVID-19 elimination in the states of New York and Florida, and the in entire US nation (albeit elimination is easier to achieve in either of the two states than in the entire US nation). Table 3 with β a = 0.8400, β s = 0.4971 for US, β a = 0.5648, β s = 0.3871 for New York and β a = 0.7675, β s = 0.4621 for Florida (corresponding to an assumed 30% reduction in the baseline value of β a and β s , to account for the effect of the implementation of social-distancing measures). 

The model (2.1) is now simulated to assess the population-level impact of the hypothetical imperfect anti-COVID-19 vaccine in the entire US and the US states of New York and Florida. The population-level impact of the vaccination rate (ξ v ) on the burden of the pandemic is monitored first of all. In these simulations, the public face mask use strategy is not implemented. The model (2.1) is then simulated using the baseline parameter values in Table 3 (with c M = ε M = 0) and various values of ξ v . The results obtained, depicted in Figure 5 , show that, for the worst-case scenario without vaccination and public mask usage (i.e., the model (2.1) with ξ v = S v = ε M = c M = 0), the US could record up to 4,000 daily deaths at the pandemic peak ( Figure 5 (a) ), while the states of New York ( Figure 5 (c)) and Florida ( Figure 5 (e)) could record up to 2,000 and 1,500 daily deaths at the pandemic peaks, respectively. The corresponding projected cumulative mortality for the entire US, the state of New York and the state of Florida, for the period February 21, 2020 to September 20, 2020, are, 120,000, 66,473 and 47,000, respectively. These simulations show a dramatic reduction in the daily and the cumulative COVID-induced mortality (in comparison to the worst-case scenario without vaccination and mask usage) with increasing values of the vaccination rate ξ v (from its baseline value). In particular, a 10-fold increase in the baseline value of ξ v for the 10 for use under a CC0 license.

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The copyright holder for this preprint this version posted May 14, 2020. . 

Other parameter values used are as given in Table 3 .

US resulted in a decrease in the daily number of deaths at the pandemic peak from 4,000 to 2,800, representing a 30% reduction ( Figure 5 (a) ). Furthermore, this corresponds to a decrease in the projected cumulative mortality from 120,000 to 98,000. Similarly, a 30-fold increase in the baseline value of ξ v resulted in a marked reduction of the daily deaths at the pandemic peak to 800 (corresponding to a decrease in the projected cumulative mortality from 120,000 to 50,000). It is evident from Figure 5 that a 10-fold increase in the baseline value of the vaccination rate (ξ v ) for the states of New York (Figures 5 (c) and (d)) and Florida (Figures 5 (e) and (f)) will bring the respective daily deaths at the peak and the projected cumulative mortality to very low levels (i.e., low levels that essentially imply the effective control or elimination of the pandemic). Thus, these simulations suggest that, while a significantly large increase in vaccination rate (from baseline) is necessarily needed to eliminate COVID-19 from the entire US, only a small increase (as low as 10%) in the baseline vaccination rate will be needed to eliminate COVID-19 in the states of New York and Florida. These simulations further show that the prospect of disease elimination using the vaccine (with the assumed protective efficacy of 80%) is more promising in the states of New York and Florida than nationwide (where much larger vaccination coverage will be needed to bring, and maintain, the control reproduction number, R c , to a value less than unity).

Simulations were also carried out to assess the impact of vaccine efficacy and coverage on this disease dynamics. This is done for the special case of the model in the absence of the use of masks in the public (i.e., the model (2.1) with c M = ε M = 0). Specifically, a contour plot of the control reproduction number (R c ) of the model, as a function of coverage (f v ) and vaccine efficacy (ε v ), is depicted in Figure 6 . This figure shows a decrease in the value of the reproduction number with increasing coverage and efficacy of the vaccine, as expected. For the assumed 80% efficacy of the hypothetical vaccine (i.e., ε v = 0.8) in Table 3 (a), the simulations depicted in Figure  6 (a) show that nationwide elimination of COVID-19 can be achieved (i.e., R c can be brought to a value below unity) if at least 90% of the US populace is vaccinated. For the state of New York, it is shown that such elimination can also occur if the state-wide vaccination coverage is at least 84% (Figure 6 (b) ). Similarly, Figure 6 (c) shows that the hypothetical anti-COVID-19 vaccine can lead to the elimination of the pandemic from the state of Florida if at least 85% of the residents of the state are vaccinated.

The effect of the combined implementation of vaccination and social-distancing strategies, on the control reproduction number of the model (2.1), is also monitored (for the US and the two US states of New York and Florida) by generating contour plots of R c , as a function of vaccine coverage and efficacy, for three levels (low, moder-11 for use under a CC0 license. This article is a US Government work. It is not subject to copyright under 17 USC 105 and is also made available (which was not certified by peer review) 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, 2020. . ate and high) of social-distancing effectiveness. The results obtained, depicted in Figure 7 , show that COVID-19 elimination (measured in terms of bringing R c to a value less than unity) is feasible even if social-distancing is implemented at a low level of effectiveness (left panels of Figure 7 ). In particular, Figure 7 (a) shows that, with the assumed 80% efficacy of the vaccine, the implementation of social-distancing at low effectiveness level will lead to the elimination of COVID-19 in the US if the nationwide vaccination coverage is at least 88%. This vaccination coverage needed to achieve such elimination decreases with increasing effectiveness of the social-distancing measures (Figures 7 (b) and (c) ). For the state of New York, for instance, vaccination program combined with social-distancing at low effectiveness level will lead to COVID-19 elimination if at least 80% of the residents of the state are vaccinated (Figure 7 (b) ). The vaccination coverage needed for elimination decrease to 67% and 61%, respectively, if the social-distancing measures are implemented at moderate (Figure 7 (e)) and high ( Figure  7 (f)) effectiveness levels. Finally, for the state of Florida, combining vaccination with social-distancing at low effectiveness level will lead to COVID-19 elimination in the state if at least 80% of the residents of the state are vaccinated (Figure 7 (g) ). The coverage needed for COVID-19 elimination in Florida decrease to 70% and 47%, respectively, for moderate (Figure 7 (h)) and high (Figure 7 (i)) effectiveness levels of the social-distancing strategy. Additional numerical simulations were carried out to assess the community-wide impact of the combined impact of vaccination with a public mask use strategy (as measured in terms of reduction in the value of control reproduction number, R c ). The results obtained, depicted in Figure 8 , show a decrease in R c with increasing vaccine efficacy and compliance. For the fixed vaccine efficacy at 80%, the minimum vaccine coverage needed to eliminate COVID-19 in the entire US if public mask use compliance is fixed at 10%, 30% and 50% is 90%, 86% and 82%, respectively (Figures 8 (a), (b) and (c)). Similarly, COVID-19 elimination can occur in the state of New York if the corresponding vaccine coverage are 82%, 80% and 71%, respectively (Figures 8 (d) , (e) and (f)). The numbers corresponding to elimination in the state of Florida are 86%, 80% and 73%, respectively (Figures 8 (g) , (h) and (i)).

Thus, the simulations in Figures 7 and 8 also emphasize the fact that the prospect of COVID-19 elimination using a vaccine is greatly enhanced if the vaccination program is complemented with another public health intervention (such as social-distancing or the use of face masks in public), particularly if the complementary intervention is implemented at moderate or high effectiveness level.

A novel Coronavirus (COVID-19) emerged in China in December 2019. The virus, which is caused by SARS-CoV-2, rapidly spread to over 210 countries causing over 4 million confirmed cases and 275,000 deaths globally (as of May 8, 2020) . Although there are currently no safe and effective vaccine for use in humans, numerous concerted global efforts are underway aimed at developing such vaccine. In fact, a number of candidate vaccines are undergoing advanced stages of clinical trials. One of the most promising of these efforts is the candidate vaccine being developed by a research group at Oxford University, which is expected to be available as early as January 2021 (or, latest, by the spring of 2021). We developed a mathematical model for assessing the the potential community-wide impact of a hypothetical imperfect vaccine against the COVID-19 pandemic in the US. The model we developed, which takes the form of a deterministic system of nonlinear differential equations, was parametrized using COVID-19 data for the entire US, as well as for the US states of New York and Florida. The hypothetical vaccine was assumed to offer imperfect protective efficacy against the acquisition of COVID-19 infection.

The model we developed was rigorously analysed to gain insight into its dynamical features. These analyses reveal that the continuum of disease-free equilibria of the epidemic model is locally-asymptotically stable whenever a certain epidemiological threshold, known as the control reproduction number (denoted by R c ), is less than unity. The epidemiological implication of this result is that the COVID-19 pandemic can be effectively controlled, whenever R c < 1, if the initial sizes of the sub-populations of the model are small enough. In other words, for this setting, the routine vaccination against COVID-19 in the US can lead to the effective control of the pandemic if the efficacy and nationwide coverage of the hypothetical vaccine are high enough. In particular, if the vaccine efficacy and coverage are able to bring (and maintain) the reproduction number, R c , to a value less than unity. Furthermore, threshold quantities for the community vaccine-induced herd immunity were calculated for the entire 12 for use under a CC0 license.

This article is a US Government work. It is not subject to copyright under 17 USC 105 and is also made available (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Table 3 , with cM = εM = 0 and various values of ξv.

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This article is a US Government work. It is not subject to copyright under 17 USC 105 and is also made available (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Table ? ? and in the absence of mask usage in public. (a) the entire US (b) the US state of New York (c) the US state of Florida. Parameter values used are as given in Table 3 with εM = cM = 0.

US and for the states of New York and Florida. We computed these thresholds for the case where social (physical)distancing is also implemented, and that the social-distancing is implemented at an effectiveness level that reduces the baseline values of the community contact rate parameters (β a and β s ) by 30%. Our computations show, using the baseline values of the parameters of the model, that at least 90% of the entire US population needs to be vaccinated in order to achieve nationwide vaccine-derived herd immunity. Similarly, at least 84% and 85% of the total susceptible population in the states of New York and Florida, respectively, need to be vaccinated to achieve such herd immunity in the states. These vaccine coverage numbers are certainly on the high side, and may not be easily attained for so many reasons (ranging from the expected insufficient stockpile of the vaccine during the early stages of their deployment, general apathy towards vaccination and the sizable percentage of people that cannot be vaccinated for medical, personal and other reasons). Our simulations showed that if the vaccination strategy is complemented with other public health interventions, such as a public mask use strategy for instance, the minimum herd immunity threshold required to effectively control the COVID-19 pandemic significantly reduces to a more realistically-attainable level. For example, if vaccination (using a vaccine with the assumed efficacy of 80%) is combined with a public face mask strategy with 30% mask compliance nationwide, the herd immunity threshold for the entire US decreases from the 90% (for the mask-free scenario described above) to 78%. Thus, the prospect of the effective control of COVID-19 using routine vaccination is enhanced if it is combined with a public mass use strategy (especially if the effectiveness and coverage of the mask use strategy are high enough).

We also carried out numerical simulations to measure the population-level impact of the rate at which the hypothetical vaccination is administered. The results obtained showed that the COVID-19 burden (as measured in terms of daily and cumulative COVID-induced mortality) decreases with increasing vaccination rate, as expected. Our simulations showed that, while significantly high increases in the vaccination rate (from its baseline value) was necessarily needed to achieve significant reduction in disease burden in the entire US, a relatively small increase in the vaccination rate from its baseline value (as low as 10% increase) will be sufficient to lead to the effective control (or even elimination) of COVID-19 pandemic in the states of New York and Florida. In other words, the prospects of eliminating COVID-19, using the hypothetical vaccine, are more promising in the states of New York and Florida, than nationwide (owing to the unrealistically high vaccine coverage needed to curtail or eliminate the pandemic in the latter).

We also simulated the combined effect of implementing the routine vaccination program with social-distancing (in the absence of face mask usage). We showed that COVID-19 can be effectively controlled using the two interventions even if the effectiveness level of social-distancing is low (as long as the vaccine coverage is high). In particular, our simulations showed that the control reproduction number (R c ) can be reduced to a value less than unity (needed for the effective control of the disease, in line with the local-asymptotic stability property of Table 3. 15 for use under a CC0 license. This article is a US Government work. It is not subject to copyright under 17 USC 105 and is also made available (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. various levels of mask compliance. (a) 10% mask compliance for the entire US (b) 30% mask compliance for the entire US (c) 50% mask compliance for the entire US (d) 10% mask compliance for the state of New York (e)30% mask compliance for the state of New York (f) 50% mask compliance for the state of New York (g) 10% mask compliance for the state of Florida (h) 30% mask compliance for the state of Florida (i) 50% mask compliance for the state of Florida. Parameter values used are as given in Table 3 with various values of cM .

for use under a CC0 license. This article is a US Government work. It is not subject to copyright under 17 USC 105 and is also made available (which was not certified by peer review) 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, 2020. . the continuum of disease-free equilibria, established in Section 3). For example, we showed that for low, moderate and high effectiveness levels for the social-distancing strategy implemented in the state of New York, COVID-19 elimination is feasible if the vaccine coverage is at least 80%, 67% and 61%, respectively (the corresponding figures for the US state of Florida are 80%, 70% and 47%). Thus, these figures suggest that COVID-19 elimination (using the combined vaccination and social-distancing strategy) is more likely in the US state of Florida, followed by the US of state of New York and then nationwide.

We assessed the population-level impact of combining the routine vaccination strategy with a public mask use strategy. Our simulations show a decrease in the burden of the COVID-19 pandemic (again as measured in terms of reduction in the value of the reproduction number R c ) with increasing efficacies and compliance of the vaccine and mask usage. It was shown that, for the case of the entire US, if the vaccine efficacy is fixed at the assumed 80% and mask compliance is fixed at the low value of 10%, COVID-19 can be eliminated in the US if the vaccine coverage is at least 90%. The vaccine coverage decreases to 86% and 82%, respectively, if mask use compliance is increased to 30% and 50% nationwide (the numbers for US state of New York, corresponding to the aforementioned mask use coverage are 82%, 80% and 71%, respectively; similarly, the numbers for the US state of Florida are 86%, 80% and 73%, respectively). Again, these result confirmed that COVID-19 elimination is more feasible is vaccination is combined with another intervention (mask use in public, in this case), and that such elimination is more easily attainable in the US states of New York and Florida than nationwide.

In summary, this study shows that the prospect of effectively controlling COVID-19 in the states of New York and Florida, as well as nationwide, is promising using the hypothetical imperfect anti-COVID-19 vaccine. The prospect of such effective control and/or elimination is greatly enhanced if vaccination is combined with other interventions that limit community transmission, such as social-distancing and mask use in public. Furthermore, elimination is more likely to be achieved using a vaccine in the states of New York and Florida than nationwide (since the latter requires much higher, and probably realistically-unattainable, vaccination coverage to achieve elimination).

for use under a CC0 license. This article is a US Government work. It is not subject to copyright under 17 USC 105 and is also made available (which was not certified by peer review) 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, 2020. . https://doi.org/10.1101/2020.05. 10.20097428 doi: medRxiv preprint 

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To mask or not to mask: Modeling the potential for face mask use by the general public to curtail the covid-19 pandemic

Scores of coronavirus vaccines are in competition -how will scientists choose the best?" a natureresearch journal (Assessed on

In race for a Coronavirus vaccine, an Oxford Group Leaps Ahead

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Transmission potential of the novel coronavirus (COVID-19) onboard the Diamond Princess Cruises Ship

Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts

Early dynamics of transmission and control of COVID-19: a mathematical modelling study

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations

An interactive web-based dashboard to track COVID-19 in real time

Coronavirus COVID-19 Global Cases by Johns Hopkins CSSE

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Coronavirus disease (COVID-2019) situation reports

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Testing the efficacy of homemade masks: would they protect in an influenza pandemic

The incubation period of coronavirus disease 2019 (covid-19) from publicly reported confirmed cases: estimation and application

Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV2)

Estimates of the severity of coronavirus disease 2019: a model-based analysis

Public health responses to covid-19 outbreaks on cruise ships-worldwide

Clinical course and risk factors for mortality of adult inpatients with COVID-19 in Wuhan, China: a retrospective cohort study

Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions

Vaccination and herd immunity to infectious diseases

The concept of herd immunity and the design of community-based immunization programmes

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One of the authors (ABG) acknowledge the support, in part, of the Simons Foundation (Award #585022) and the National Science Foundation (Award #1917512). CNN acknowledges the support of the Simons Foundation (Award #627346).