key: cord-0308973-ezqqbvq7
authors: Roy, J.; Heath, S.; Ramkrishna, D.; Wang, S.
title: Modeling of COVID-19 Transmission Dynamics on US Population: Inter-transfer Infection in Age Groups, Mutant Variants, and Vaccination Strategies
date: 2021-09-28
journal: nan
DOI: 10.1101/2021.09.25.21264118
sha: 1999c6bf3445f84d53d3cdd85b6d0b852b8a26b7
doc_id: 308973
cord_uid: ezqqbvq7

The in-depth understanding of the dynamics of COVID-19 transmission among different age groups is of great interest for governments and health authorities so that strategies can be devised to reduce the pandemic's detrimental effects. We developed the SIRDV-Virulence epidemiological model based on a population balance equation to study the effect of mutants of the virus and the effect of vaccination strategies on mitigating the transmission among the population in the United States. Based on the available data from the Centers for Disease Control and Prevention (CDC), we obtain the key parameters governing the dynamic evolution of the spread of the COVID-19 pandemic. In the context studied, the results show that a large fraction of infected cases comes from the adult and children populations in the presence of a mutant variant of COVID-19 with high infection rates. We further investigate the optimum vaccine distribution strategy among different age groups. Given the current situation in the United States, the results show that prioritizing children and adult vaccinations over that of seniors can contain the spread of the active cases, thereby preventing the healthcare system from being overwhelmed and minimizing subsequent deaths. The model suggests that the only option to curb the effects of this pandemic is to reduce the population of unvaccinated individuals. A higher fraction of 'Anti/Non-vaxxers' can lead to the resurgence of the pandemic.

The coronavirus disease of 2019 (COVID- 19) pandemic has been ravaging the entire 2 world ever since its inception in Wuhan, China back in December 2019 [1] . The most 3 challenging part of analyzing and predicting the COVID-19 pandemic in terms of 4 infection and mortality numbers is its ever-changing dynamics due to the mutations of 5 the virus [2] . For COVID-19, its mutations often cause an increase in the transmission 6 rate of the virus [3] due to changes in its spike protein [2] . These mutations strengthen 7 the interaction between COVID-19 virus and its receptor ACE2 [4] . This in turn also 8 affects the vaccine efficacy [5] and thus creates a possibility of resurgence of the 9 pandemic [6] , which is revealed from the changing trends of the active infected cases 10 and deaths in the US in recent times [7] . Since their peak in early January 2021, the 11 COVID-19 cases and deaths had markedly declined, due in part to the increased 12 vaccination coverage [8] . However, during June 19-July 23, 2021, COVID-19 cases 13 increased approximately 30% nationally, followed by increases in hospitalizations and 14 deaths [8] , driven by the highly transmissible B.1.617.2 (Delta) variant, a variant of the 15 severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). 16 The Delta variant is more than two times as transmissible as the original strains 17 circulating at the start of the pandemic [9] and is causing large, rapid increases in 18 infections, which puts pressure on the local and regional health care systems to provide 19 medical care [9] . Strains on critical care capacity can increase COVID-19 20 mortality [10, 11] while decreasing the availability and use of health care resources for 21 non-COVID-19 related medical care [12, 13] . In this scenario, it is important to predict 22 the timing of peaks for active infections regarding worst-case scenarios, so that medical 23 personnel are prepared and the Government can introduce appropriate and informed 24 policy decisions. 25 One of the primary concerns is to identify the age groups which will be affected the 26 most by delta strain of SARS-CoV-2 [14] . Age-related differences in responsiveness and 27 tolerance become obvious and lead to worse clinical outcomes in elderly individuals [15] . 28 Previous studies have mentioned that older COVID-19 patients are at an increased risk 29 of death [16] [17] [18] [19] . Accordingly, optimum vaccine strategies need to be developed, 30 prioritizing the most affected age groups to reduce the peak of infected cases. 31 Despite widespread availability, vaccine uptake in the US has slowed nationally with 32 wide variation in coverage by state (range = 33.9%-67.2%) and by county (range = The COVID-19 disease has affected different age groups to different extents. Studies 48 by Chikina et al. have developed SIR-like epidemic models integrating known 49 age-contact patterns for the United States to model the effect of age-targeted mitigation 50 strategies for a COVID-19-like epidemic [25] . It has been seen that strict age-targeted 51 mitigation strategies have the potential to greatly reduce mortalities and ICU 52 utilization. Another study by Yu et al. reaffirmed that COVID-19 epidemic processes 53 have had distinctive dynamic patterns among age and gender group [26] . The epidemic 54 among young adults led the epidemic process across the whole population, with a 55 second peak occurring in people aged 20-39 years. Studies by Moghadas et al. have 56 highlighted the importance of incorporating mutations and evolutionary adaptations in 57 epidemic models [27] . They also demonstrated how the multiple-strain transmission 58 model can be used to assess the effectiveness of mask-wearing in limiting the spread of 59 COVID-19. Thus, in this scenario vaccination strategies become of primary importance. 60 Vaccination markedly reduced adverse outcomes, with non-ICU hospitalizations, ICU 61 hospitalizations, and deaths decreasing by 63.5% (95% CrI: 60.3% -66.7%), 65.6% (95% 62 CrI: 62.2% -68.6%), and 69.3% (95% CrI: 65.5% -73.1%) [28] , respectively, across the 63 same period. To date, there is an urgent need for studies focusing on the effects of 64 different age groups on the COVID-19 transmission dynamics with mutual 65 considerations of mutation and vaccination strategy. In this work, we model the 66 interaction among the age groups as the infection brought by the virulence 67 environment [29, 30] . We correlate factors of variants and vaccination efficacy by 68 studying the effect of mutants on various age groups and devise vaccine distribution 69 strategies. 70 In this study, we will apply the population balance model to derive the average 71 equation, which is used to simulate the COVID-19 transmission that has occurred in the 72 United States. We will focus on three aspects of COVID-19 transmission modeling, as 73 illustrated in figure 1(a) . Firstly, we will evaluate how an increased transmission rate of 74 delta variant and the increased vaccine inefficacy due to the mutation can change the 75 dynamics of the pandemics in the US. We will identify the age groups which are 76 supposed to be affected the most by this new strain. Secondly, we will attempt to 77 design an optimum vaccine distribution strategy prioritizing the most affected age 78 groups so as to bring down the active infections and mortality. Thirdly, we will account 79 for the effect of anti-vaxxers to determine what proportion of population needs to be 80 vaccinated to prevent the resurgence of the pandemic.

Our model considers the interactions among different age groups using population 83 balance equations [31, 32] . The subsequent average equations are fitted under the 84 collected COVID-19 transmission data. In the following sections, we will start with the 85 introduction of the model for the infection transfer among three different age groups (i.e. 86 children, adult, senior); then derive the average equations using the population balance 87 model, which can be converted to dimensionless equations with characteristic quantities 88 (dimensionless numbers); we quantify the dynamic of these dimensionless numbers using 89 CDC released infection, death, and vaccination data. Average infection inter-transfer frequency between any two age groups -98 Here, we identify the age-specific infection transfer frequency as q(τ, τ ; t)dt in figure   99 1(b), which is the probability that two age groups τ and τ who are in each other's 100 neighborhood will meet with the other in closed proximity during the time interval t to 101 t + dt. For instance, we choose τ to be in the children range denoted as g = S 1 (S: 102 Susceptible) and τ to be any other age range (e.g. adult, elderly) identified as 103 g ∈ {I 2 , I 3 } or self g = I 1 (I: Infected). The average meeting frequency between 104 individuals in g and g is identified as: coordinates where v denotes the viral population in the individual and a represents the 110 antibody population. Thus, we let φ(v, a, t)dvda be the number of individuals with a 111 viral infection in the range v to v + dv and an antibody in the range of a to a + da. The 112 viral infection in each individual changes at the averaged rateV i in population i, where 113 V(t) = v dv a daφ(v, a, t) is denoted as virulence. Note that, in this work, we focus on 114 the effect of virulence environment among population interactions and neglect the factor 115 of viruses resident on the solid surfaces and droplets [33] . Therefore, this definition of 116 virulence V excludes viruses resident on the solid surfaces in the environment, which can 117 also cause spreading to the extent that people come into contact surfaces. 118 We note the total population density of infected individuals I i ,

where a i indicates the age range of group i; we assume that a freshly infected individual 120 has v = 0 and a = 0. Then,V i (v) is the viral infection change rate in the individual and 121 A i (a) is the antibody concentration change rate in the individual. An infected individual 122 with viral intensity v may die with the frequency k d (v, a) or recover with frequency 123 k r (v, a). The population balance equation for the diseased population is given by

We denote the existence of probability densities p i (v) in age group i for developing an 125 infection on contact with an infected individual of viral density v, All rights reserved. No reuse allowed without permission. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. in the United States, in which the members of three distinct age groups move to compartments (blue) at the rates given adjacent to the inter-compartmental arrows (blue). We consider the infection inter-transfer frequency among three different age groups (children, adults, and seniors). Infected members of each age group contribute to a single virulence parameter (orange), which can infect both susceptible, (S i ), and vaccinated, (V i ), individuals. b) Data is fed to the compartmental model to fit its parameters. The fitted model is used to run simulations to predict future scenarios: 1) the effect of the delta variant, 2) the effect of changing the vaccination allocation and the vaccination roll-out speed, and 3) the effect of changing the proportion of COVID-19 Anti/Non-Vaxxers. boundary condition for equation (3) is (4) where S i is the susceptible population, V i is the vaccinated population, andq Si−Ij is the 128 meeting frequency between susceptible S i and infected I j .

Average Models -We further observe that the total infection equation obtained by 130 integrating equation (3) with respect to v from zero to infinity and a within the age 131 range (a i ),

where we applied the boundary condition equation (4) andĀ(0) = 0; the transmission 133 rate β i and vaccine efficacy σ are thus defined as

The rate constant β characterizes the extent of the inter-transmission among different 136 age groups, and σ measures the ratio of transmission rate between vaccinated and 137 susceptible populations. The death and recovery rates are expressed as

Next, we multiply equation (3) by v and integrate with respect to v and a from 0 to 140 infinity:

With dynamics of virulence V and infection I i , it would be straightforward to derive the 143 equations for the rest of compartment, which has been displayed in supplementary

Time to infect an individual × Rate of viral multiplication relative to v 0 and N e was abstracted as its dimensionless form:

where a list of dimensionless parameters is defined in Table 1 . The susceptible fraction 149 for age group i is represented by x S,i ; in figure 1 (b) and equation (12) preprint (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 this version posted September 28, 2021. ;  governed by dimensionless recovery rate (b i ) and dimensionless mortality rate (c i ), 162 respectively (see equations (14) - (15)). In equation 16, the population in the vaccinated 163 compartment is governed by the vaccination rate and the vaccine efficacy. Lastly, the 164 dynamics of virulence are governed by the dimensionless number d i , which is 165 contributed by the infected compartment. There is also a removal term from the viral 166 load compartment dictated by dimensionless number e, which is related to the lifespan 167 of the virus.

Data Processing -We considered three different age groups for analysis: children 169 (12-17 years), adults (18-64 years), and seniors (65 years and older). These age group 170 divisions mirrored those of the CDC's vaccination data set [35] . Because the infected 171 cases and death data sets used for this study had smaller age group divisions than those 172 of the vaccination data set, the age group data of each data set were combined to form 173 uniform age groups across all three data sets (figs. S1-3 in supplementary material). To 174 analyze the COVID-19 dynamic evolution, we quantified dimensionless quantities based 175 on data of infection, death and vaccination, which was normalized by the total 176 population. For the analysis, the total population of the United States was assumed to 177 be 332.5 million [36] , based on the current US population at the time of data collection. 178 The United States Census 2019 age distribution [36] was used to estimate the 179 population of age groups in the United States. Because each data set used in this study 180 had a unique format, the data sorting and processing for each set were different, as 181 described in the supplementary material.

Simulation Scheme -Before fitting the dynamic SIRDV-Virulence equations 183 (12)- (17) to the case, death, and vaccination data, it was necessary to estimate the 184 initial value for each compartment. The method for estimating the initial value of each 185 compartment has been outlined in supplementary materials, which was repeated for 186 each of the three age groups.

Note that data were not available for the susceptible population. Since the 188 Susceptible compartment is the final compartment under consideration, a population 189 balance was used to calculate this value:

where N i is the population of age group i and N is the total population (ages 12 and 191 older). For simplicity, it is assumed that the population of each age group is constant 

The dynamics of vaccination rate a i correctly captures the current vaccination 240 strategy in the United States. During the first two time periods, a i was higher for the 241 senior age group (i = 3), which is consistent with prioritizing senior vaccination. In the 242 last two periods (figure 2e), the children and adult age group, on average, have a higher 243 vaccination rate than the senior age group.

The recovery rates for all the three age groups shows an increasing trend among all 245 three age group as time progresses. This is indicated in figure 2f . The main reason 246 behind this, with a higher fraction of the population getting vaccinated with time, the 247 immunity is expected to increase. However, there is an increase in the mortality rate of 248 the senior age group from the third to the fourth time period as indicated by a high 249 value of c in figure 2g, which can primarily be due to the delta variant effect. Relative 250 to the senior age group, the changes in mortality rates of children and adults are preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. In the following sections, we will discuss the future predictions considering three The mutation of the virus has been largely responsible for the increase of 262 transmissibility due to increased infection rate and reduced vaccine efficacy [40] . 263 Recently, the delta variant is responsible for almost all recorded COVID-19 cases [7] , in 264 accordance with the elevation of infection (K i ) in the fourth fitted time period. Further simulation scenarios can be found in the supplemental material: variation of 292 K with σ = 0.20 (figs. S12-13), variation of σ with K new /K = 1 (figs. S14-15), and 293 variation of σ with K new /K = 2 (figs. S16-17).

Vaccination optimization strategy 295 Optimization of vaccine distribution strategies among different age groups remains 296 critical [41] . Specifically, it would be interesting to study the effect of varying the 297 vaccination rates and vaccination prioritization among each of the three age groups.

Our study modeled the resulting completed vaccinations, cumulative cases, and 299 cumulative deaths over a future time period as the vaccination parameters were varied. 300 To determine the practical range for the dimensionless vaccination parameter a i of 301 each age group, we need to consider two constraints. Firstly, the maximum completed 302 vaccination rate for future simulations for a single age group was set as the maximum 303 weekly completed vaccination rate achieved during the fitted time period for the 304 respective age group ( fig. S5 in supplementary material) . Secondly, the maximum 305 completed vaccination rate for future simulations for all age groups combined was then 306 September 25, 2021 12/24

All rights reserved. No reuse allowed without permission. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. set as the maximum weekly completed vaccination rate during the fitted time period. In 307 other words, the predicted simulations were completed under the assumption that 308 future vaccination rates in the United States would not reach the rates that they had 309 reached previously (considering both individual age groups and the entire population 310 under study), given the majority of the senior and adult age groups had already been 311 vaccinated by the initial date of the future simulation time period and peaks had 312 already been reached for the completed vaccination rates in each age group.

In figure 4 , a comparison was done for vaccination priority for age groups with the 314 heat maps. In general, the minimum infected cases and deaths and the maximum 315 fraction of vaccinated population occur at the highest values of a i for these age groups, 316 yet the dependence of vaccinate rate in each age group is different. For instance, the 317 future total infected and deaths, as well as total vaccinated fraction are more dependent 318 on a 2 (adult age group) than on a 1 (children age group) as shown in figures 4(a,b) . This 319 is mainly because the fraction of the adult population is much higher than that of 320 children. For the comparison among children and senior age groups, it was seen that the 321 total death and infection were more strongly dependent on a 1 than on a 3 (senior age 322 group) as seen in figures 4(d,e) . This is because a large fraction of seniors (∼ 323 81.8%) [42] had already been fully vaccinated for COVID-19. In comparison, only 324 around 54.4% of the adult population was vaccinated by the end of July 2021, while the 325 percentage of children fully vaccinated was even lower (∼ 34.4%) [42] .

Since a large fraction of children population is yet to be vaccinated, a higher priority 327 needs to be given to the children age group over the senior age group for the future 328 vaccine distribution, which is consistent with the current ongoing strategy in the United 329 States [43]. A similar comparison among the senior and adult age group showed that 330 total death and infected cases is more dependent on a 2 than a 3 as seen in figures 4(g,h) . 331 The reason is same as the previous case: a large fraction of adult category had yet to be 332 vaccinated by the end of July 2021. So the observations from these heat maps led to the 333 conclusion that, for the vaccine distribution strategy, a higher priority needs to be given 334 to adult and children categories than senior category to minimize total death and 335 infection cases as the majority of population in these two groups are more susceptible to 336 the infection. 

The above differential equations describe the dimensionless behavior of each 

As shown in figure 5(b) , an increase in the proportion of Anti/Non-Vaxxers will lead 373 to higher virulence and in turn a higher number of infected cases and deaths. The 374 higher the fraction of vaccinated people, the lesser will be the number of deaths and 375 infected cases due to a lower virulence. The future simulations were run in sets, first 376 varying ω i for each age group while keeping that of the other age groups constant. For 377 these simulations, when ω i was varied for a single age group, i, the dynamics of age 378 group i had significant changes, but negligible changes in the dynamics of other age The PAIRDV-Virulence model, a variation of the SIRDV-Virulence model, is introduced, in which the susceptible population, S i , is divided into a sub-population that will not be vaccinated, A i , and a sub-population that will be vaccinated, P i . (b) When the proportion of the susceptible population that will not be vaccinated, ω i , increases, the virulence, infections, and deaths increase(c,d,e,f,g,h). The predicted cases and deaths are shown for (c,f) seniors, (d,g) adults, and (e,h) children as a result of simultaneously changing ω i for each age group.

considered, the adult and children seem to be affected more than the senior age group. 391 This is justified by the fact a large fraction of senior population has already either been 392 infected or vaccinated, relative to the other two age groups.

In case of unchanged transmission rate of the virus and vaccine efficacy, the peak in 394 active infection is expected to occur sometime in October 2021, whereas if the 395 transmission rate K i doubles and the vaccine efficacy 1 − σ falls to 0.8, the number of 396 active infections at the peak is expected to go higher by about two times and is 397 expected to occur sometime in December 2021. Although there will be a slight increase 398 in total deaths in this worst case scenario, it will not be a significant one, supporting 399 the point that vaccines indeed bring down the mortality rate of COVID-19. seniors, priority to adults, and priority to children. It was seen that the total 413 vaccination and total infected cases seemed to be heavily dependent on both relative 414 dimensionless a 1 (children) and a 2 (adults) as compared to a 3 (seniors). A higher value 415 of relative dimensionless parameter a i suggests a higher proportion of vaccines being 416 administered to that group. The deaths however were not affected much by a 3 and 417 predominantly affected by a 1 and a 2 . These conclusions seem logical as a large fraction 418 of adults and children are still left to be vaccinated and any changes in vaccination rate 419 for these two compartments will affect the overall vaccination and infection. Also the 420 children do not contribute much to the death compartment due to very low mortality 421 rate and hence changes in a 1 won't affect the total deaths which are heavily dominated 422 by adults and seniors. Since the adult age group consists of the greatest proportion of 423 the US population, any change in vaccination rate for adults will have a drastic effect 424 on the overall dynamics. Thus it can be concluded from the heat maps that prioritizing 425 the adults and children over seniors for vaccination will be a more effective approach for 426 minimizing the future infected cases and deaths and maximizing the fraction of With the SARS-CoV-2 virus undergoing mutations at a fast rate, it becomes of 440 immense importance to study the effect of these mutants on the transmissibility rate 441 and vaccine dynamics. While different vaccine manufacturers make different claims 442 about the effectiveness against the mutants, the exact effect of these mutants on vaccine 443 efficacy is still not clear [2] . In such a scenario, it becomes important to predict the 444 effect of the variation on the population, considering increased infection transmissibility 445 and decreased vaccine efficacy to predict the worst case scenarios for future in terms of 446 active infections and total deaths. The predictions of our studies not only help to 447 predict the peak time of infection and help the healthcare system to prepare accordingly 448 but also assist in devising the optimum vaccination strategy, prioritizing the children 449 and adult age groups, which seem to be affected the most by the delta variant.

Along with the introduction of the new variant, another aspect of primary concern is 451 that a certain fraction of the population is not willing to get vaccinated. As the novelty of the study lies in two parts: First, it identifies the age groups which are likely 458 to be worst affected by the delta variant and suggests the optimum vaccine distribution 459 strategy. Secondly, it re-emphasizes the fact that a large fraction of unvaccinated people 460 has catastrophic effects and can cause recurrence of the pandemic. For future studies, it 461 will be interesting to include the dynamics involved with the mutation of the 462 SARS-CoV-2 virus and study how the changing mutation rate can affect the present 463 vaccines being administered. Also, it will be beneficial to identify the specific 464 geographical regions which are likely to get affected the most by the delta variant, 465 taking into account the age distribution in that state and the status of vaccination.

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