key: cord-0906327-yc8jdxm0
authors: Topîrceanu, Alexandru
title: Immunization using a heterogeneous geo-spatial population model: A qualitative perspective on COVID-19 vaccination strategies
date: 2021-12-31
journal: Procedia Computer Science
DOI: 10.1016/j.procs.2021.08.217
sha: 0b5af0bc1a1c44b6c5d47eeaf278238120301968
doc_id: 906327
cord_uid: yc8jdxm0

Epidemic modeling has been a key tool for understanding the impact of global viral outbreaks for over two decades. Recent developments of the COVID-19 pandemic have accelerated research using compartmental models, like SI, SIR, SEIR, with their appropriate modifications. However, there is a large body of recent research consolidated on homogeneous population mixing models, which are known to offer reduced tractability, and render conclusions hard to quantify. As such, based on our recent work, introducing the heterogeneous geo-spatial mobility population model (GPM), we adapt a modified SIR-V (susceptible-infected-recovered-vaccinated) epidemic model which embodies the idea of patient relapse from R back to S, vaccination of R and S patients (reducing their infectiousness), thus altering the infectiousness of V patients (from λn to λr). Simulation results spanning over a period of t = 2000 days (6 years, the period « 2020-2025) compare the impact of an epidemic outbreak with variable vaccination strategies, starting after 1 year (as is the case of COVID-19). The infected proportion in the remaining 5-year period is analyzed using vaccination rates from rv = 0 (no vaccination) to rv = 1. While rv < 0.4 is less effective during the earlier stages, all strategies with rv > 0.4 show a similar downward convergence reducing the number of infected by more than half, compared to no vaccination. Given the complexity of epidemic processes, we conclude that higher vaccination rates yield similar results, but a minimal rv = 0.4 (40% of population over five years) should be targeted.

Many recent studies in epidemic modeling are predominantly developing mass-action models into tools suitable for analyzing COVID-19 dynamics [9, 11, 4, 15, 18, 12] . Too often, nevertheless, these studies focus on customizing the underlying epidemic models (e.g., SIR, SEIR, SIRS), but consider homogeneous mixing of the population, implying that all individuals are fully connected inside a compartment or stochastic block [6, 27, 5, 4, 12] . Some studies propose

Many recent studies in epidemic modeling are predominantly developing mass-action models into tools suitable for analyzing COVID-19 dynamics [9, 11, 4, 15, 18, 12] . Too often, nevertheless, these studies focus on customizing the underlying epidemic models (e.g., SIR, SEIR, SIRS), but consider homogeneous mixing of the population, implying that all individuals are fully connected inside a compartment or stochastic block [6, 27, 5, 4, 12] . Some studies propose so-called "flattening the curve" solutions that try to reduce the reproduction number R 0 , but are also underpinned by homogeneous mixing models. Conversely, Watts et al. [36] found R 0 to have little influence on the predictability of large-scale epidemics, in the context of real-world, multi-scale population structures. Real-world population models are dictated by geographic, as well as historical, demographic and economic factors. As such, simplified population (and mobility) models, which lack the complexity of global scale population organization can lead to over-or underestimations in terms of epidemic size and impact [8, 17, 12] , or outbreak duration [4, 6, 18, 15] .

Indeed, several state-of-the-art contributions developed increasingly realistic epidemic models by incorporating heterogeneous population mixing and human mobility patterns. Using Network Science, Salathé et al. [23] found that the topology of networks is a critical detail in explaining infectious dynamics, based on empirical data for viral diseases, e.g., SARS [20] , HIV [3] , measles [13] , and influenza [12] . Furthermore, community structure has been observed to dictate the speed of epidemics. To this end, Shang et al. [25] study how overlapping communities and a higher network average degree accelerate infectious spreading. Salathé et al. [23] explain how the dynamics of epidemics over communities have implications on immunization strategies for large outbreaks. Chen et al. [7] show how overlapping in communities leads to increased infection rates and a higher spreading speed. Finally, we note the work of Stegehuis et al. [26] who demonstrate that the mesoscopic level of communities influences the behavior of percolation on networks, by stimulating or suppressing spreading speeds.

The main goal of this paper is to offer a qualitative forecast of vaccination strategies' efficacy, in the context of COVID-19. To this end, we employ epidemic modeling using an original heterogeneous population mixing model supported by geo-demographic data and daily estimations of COVID-19 cases. In our recent study [28] , we introduce the geo-spatial mobility population model (GPM) which was compared to analogous homogeneous mixing models, and validated against real COVID-19 data [10] . Nevertheless, in [28] , our research focus was on the community structure and individual mobility rather than on the transmission model; hence, a standard SIR epidemic model was incorporated into GPM to run numerical simulations. By contrast, this paper uses the underlying population mobility model of GPM and adapts a SIR-V (susceptible-infected-recovered-vaccinated) epidemic model with several considerations meant to increase the realism and the reliability of our analysis:

1. All recovered patients relapse back to susceptible after an average timeout of 3-6 months, as defined by recent studies [34, 14, 37] . Consequently, the recovered state is not a "final" state, as in the case of a classic SIR model. 2. All susceptible and recovered individuals can undergo vaccination defined by a constant rate r v . During each simulation iteration, a constant proportion of random non-infected individuals are selected for vaccination. 3. Transmission to the vaccinated state will not render individuals immune to the epidemic. Conversely, as recent studies suggest [2] , vaccinated individuals are less likely to get infected by roughly 75%. In other words, all vaccinated susceptible individuals are transformed back into susceptible individuals with a reduced infection probability λ r , as opposed to non-vaccinated (normal) susceptibles with λ n . Vaccinated recovered individuals keep their recovered state until the relapse timeout, when they become susceptibles with the reduced infection rate λ r .

Given the increased realism of the GPM population model [28] and the proposed SIR-V epidemic model, inspired by recent scientific findings, we study the effect a variable vaccination rate r v over a moderate time window (5 years). We reproduce a GPM model for several countries with different population densities, then analyze the dynamics of infected and recovered proportions using computer simulation.

Settlements are the basic form of human organization as they delimit the boundary between inhabited and noninhabited land. To simplify our assumptions, the GPM model considers two levels of population organization, namely that of countries, and settlements (i.e., villages, towns, cities, regardless of region and size) inside each country. The current COVID-19 pandemic has shown that countries, even though an artificial political construct, are a highly relevant form of population organization, as they can adopt different policies for population mobility (including lockdowns, social distancing, mobility restrictions) further influencing epidemic dynamics [33, 19] .

The first ingredient of GPM is the generated realistic population structure. In essence, GPM translates into a two dimensional mesh of stochastic block models [16] , with each block corresponding to a human settlement s i (defined by a given country). Each settlement s i is characterized by its real-world geographical position, using latitude (y) and longitude (x), and its size Ω i (i.e., number of inhabitants). For higher accuracy, we use GPM at a country level, rather than a continental or global level, due to the sparsity of available data. All individuals from a settlement n j ∈ s i are considered as interconnected, thus similar to a SIR compartmental model applied independently for each settlement. Nevertheless, the division of a total population into multiple (e.g., hundreds, thousands) of unique blocks is a significant factor favoring the realism of epidemic dynamics.

The second important ingredient of GPM is the modeled individual mobility. All individuals from a settlement s i have the same probability of leaving to another settlement s j . As such, we express the normalized probability of mobility from a settlement s i to s j as p(s i , s j ), given by the following equation:

where Σ k p(s i , s k ) = 1, Ω j is the size |s j | of settlement s j , d i j is the geographical distance (i.e., Euclidean distance (∆x 2 + ∆y 2 ) 1/2 ) between the two settlements s i and s j , and ψ is a tunable parameter. The ψ parameter can be used to amplify or reduce global mobility; we use a fixed ψ = 0.2 in this study, translating to the reduced mobility seen throughout the COVID-19 period (2020-2021). For a full definition of the geo-spatial population model (GPM), which is beyond the goal of our current paper, please refer to our previous contribution [28] .

The proposed epidemic model is summarized in Figure 1 which defines the four states and particular transition rates. Any individual can be in one of the four states: susceptible (S), infected (I), recovered (R) or vaccinated (V). The transition S → I → R is determined by the classic (normal) infection rate λ n , and recovery rate µ, respectively. For these transition rates we chose the values λ n = 0.06 and µ = 0.02, supported by recent studies which measured an R 0 = λ n /µ = 3.3 ± 1.4 for COVID-19 [1] . Fig. 1 . Overview of our modified SIR epidemic model into a SIR-V model with the following states: susceptible (S), infected (I), recovered (R), vaccinated (V), and specific transmission rates: normal infection rate λ n , infection rate for vaccinated individuals λ v , recovery rate µ, vaccination rate r v , and average timeout for patient relapse δ r . We consider V a transitory state (hence the dotted contour) since all S/R individuals undergoing vaccination return to their previous S/R state, with the new (lower) infection rate.

Unlike the classic SIR model, we do not model R as a final state for individuals. Given the relatively short immunity to the SARS-CoV-2 virus, all individuals in R will eventual transition back to S after a relapse (timeout) period δ r . The relapse period for any recovered individual is uniformly chosen between of 3-6 months (i.e., 90-180 iterations), supported by recent estimates of COVID-19 immunity [34] (3 months), [14] (4-5 months), and [37] (6 months) .

Also, any susceptible and recovered individuals can undergo vaccination with a rate of r v (estimated for a 5 year period). For simplicity, we interpret r v as a global "successful" vaccination rate, not considering the different types of vaccines currently available. From r v , a daily vaccination rate d v is computed as d v = r v /5/365. Thus, a proportion d v of randomly chosen, non-infected and non-vaccinated individuals transition to the V state every iteration (day). However, this state is transitory, as it simply alters the rate at which nodes will eventually infect and become infected by others. All nodes in V are marked in such a way that their new infection rate becomes λ v . Vaccination performance is known to vary between 60-95% (based on existing products), however, we use λ r = 0.25 · λ n , supported by recent research findings, published in The Lancet, that the reduction in COVID-19 cases dropped by 75% (down to one quarter) compared to non-vaccinated individuals [2] .

Overall, our proposed SIR-V model implies a continuous transition of nodes between states S → I → R . . . S , defined by the relapse timeout δ r , and the reduction of λ n to λ v (used both for infecting and being infected) after each node is vaccinated.

We run GPM at country level, i.e., modeling all settlements inside a chosen country. As such, the number of settlements (and the total simulation population Ω = i Ω i , are defined by a chosen geographical area A, limited by a country's borders. Furthermore, the position (x, y) and size Ω i of settlements s i are defined in accordance to the Global Rural-Urban Mapping Project (GRUMP v1) curated by the Center for International Earth Science Information Network (CIESIN), Columbia University [35] . Since the project is still evolving, it is missing some data for various regions of the world. Nevertheless, we choose to model three countries, representative in terms of population density (i.e., which in turn influences the amount of human mobility). In this paper we run a set of experiments on models of Romania, Germany and South Korea. These countries were chosen because of their increasing population densities, as detailed in Table 1 , which summarizes the demographic data of each used country model. The focus is to show how population and density play a role in the epidemic dynamics. Table 1 . Country models used in this study, detailing their population density, the real country size (as defined by the GRUMP dataset), the simulation population (used by GPM), and the number of settlements considered in GPM. We note that, in order to speed-up the simulation, without loosing notable predictive accuracy [28] , we reduce the modeled population Ω uniformly, from reach settlement, by a factor of ×1000 (see column "Simulation size (GPM)" in Table 1 ). To keep the same population density, we also reduce the size of the total area A by the same ×1000 factor. Experiments show that, as long as the density (=Ω/A) is kept constant, by reducing both Ω and A, the dynamics of the modeled epidemic maintain the same characteristics.

In order to compare our computer simulations output with real epidemic data, we use a recent snapshot of the JHU CSSE COVID-19 dataset, that is being curated by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University [10] . This comprehensive dataset contains time series information on daily accumulated confirmed Coronavirus cases worldwide. From these numbers we express the absolute daily number of cases, as the difference between each two consecutive days. Also, we estimate the number of infected individuals in a country, for any day d, as the sum of daily cases over the two weeks prior to d (i.e., the sum of daily cases from d − 13 to d). This simplification of the estimation of total infected cases is [21] .

In this section we present the summary of numerical simulations on three country models, with population structure and mobility defined by GPM (see Table 1 for details), and epidemic dynamics according to the proposed SIR-V model (see Figure 1 for details). Our goal is to provide a qualitative assessment of the performance of vaccination strategies. As such, we try to understand how the proportions of infected and recovered individuals (normalized by country population, and expressed in % for convenience) vary based on an increasing vaccination rate r v = 0, 0, 1 . . . 0.9, 1.

All simulations run for a fixed duration of t = 2000 iterations (i.e., days), with vaccination beginning after one year (t = 365), as is the case of COVID-19, offering a 5-year post-vaccination overview of the epidemic. For simplicity, we note that the geo-spatial data and the population model do not evolve during this time. Figure 2a shows the dynamics of COVID-19 using real data from [10] in Romania, measured by the total infected proportion, estimated from the daily number of cases. Here we notice three irregular waves (w 1 − w 3 ) occurring at non-deterministic moments in time. A similar dynamic characteristic is observed in Figure 2b (red dotted area) where the simulated dynamics for Romania (blue) are very similar. Additionally, Figure 2b also plots simulation dynamics for Germany (orange) and South Korea (green). The simulation scenario employs a moderate vaccination rate of r v = 0.3. We notice high variability in the early stages of the simulation (t = 0 − 400), that suggest an association with the population density. Indeed, South Korea did adopt early isolation strategies, so it did not have the same infection spike as depicted in Figure 2b (green). After ≈ 400 days, the infected proportion has a declining convergent trend, regardless of country density. The same conclusion can be drawn from the inset of Figure 2b , where the recovered proportion converges for all countries. In Figure 3 we depict the impact of vaccinating the general population by comparing three strategies: no vaccination (r v = 0), moderate vaccination (r v = 0.3), and strong vaccination (r v = 0.9) over a 5-year period (Jan/2021-Dec/2025). It is obvious that the effects of the pre-vaccination period cannot be undone, but once vaccination starts (t ≥ 365), it becomes clear that both strategies (blue, green) reduce the number of infected individuals noticeably. During the early vaccination period (2021-2022), infection waves will still be probable for the moderate strategy (r v = 0.3), while the stronger strategy seems to dampen these waves better. During the moderate-term period (2023-2025), we notice that both vaccination strategies converge similarly, and keep the infected ratio much smaller than without any vaccination. Specifically, our SIR-V model pinpoints a drop in the infected proportion from > 0.4% (orange line) in case of no vaccination, down to < 0.2% infected in case of any of the two other vaccination strategies (green, blue lines).

Next, we focus on detailing the impact of a more granular vaccination rate r v . Figure 4a depicts a snapshot from t = 1000 − 2000 (roughly Jan/2023-Dec/2025), after the infected proportion has dampened for most vaccination Fig. 3 . The impact of a moderate vaccination strategy (30%, green line), a strong vaccination strategy (90%, blue line) versus no vaccination in the population (0%, orange line). The first part (pre-vaccine) of the simulation period is more unpredictable due to the heterogeneity of population structure and mobility (considered by our GPM model). After ≈ t > 400 we notice a significant impact of both vaccination strategies -larger waves are avoided, and the dynamics converges towards a smaller overall infected proportion of < 0.2%, instead of > 0.4% visible for the no vaccination strategy. Nevertheless, our simulation shows no significant difference between the 30% and 90% strategies given a moderate temporal perspective (2-5 years). strategies, which suggest no more visible epidemic waves. Without a vaccination policy (r v = 0, dark blue line), the infected count would reach a high, pseudo-constant proportion of 0.4-0.5%, comparable to a permanent infection wave. To put the simulation results into perspective, in early 2021, Romania saw a second wave ending and a third wave unfolding with about 0.3-0.5% of the population infected with SARS-CoV-2.

The "weakest" vaccination strategy, depicted in Figure 4a , is r v = 0.2 (orange line) and converges slowly towards the dynamical characteristic of the other stronger strategies (r v ≥ 0.4). That is, all other strategies with r v ≥ 0.4 produce similar infection dynamics. We note that the pandemic is not eradicated over the 5-year simulation window, but it remains as a permanent viral outbreak without significant waves. We believe this happens because the vaccination does not render total immunity (rather 75% reduction in ineffectiveness), and because the global vaccination is most likely going to be a very slow process. Supporting the same observations is Figure 4b , which depicts the recovered proportion for all vaccination rates, during the whole simulation period t = 0 − 2000. Note that the recovered population does not grow constantly, as in the case of SIR, because our SIR-V epidemic model implies a continuous relapse from R → S . Plotting the recovered proportion gives clearer insights into the efficacy of vaccination strategies. To that end, by focusing on the early to mid vaccination period (t > 400), we can consider only the rates r v ≥ 0.4 as effective, in terms of keeping the recovered proportion below 3% at any time. When r v = 0.2 (orange line) we measure a duration of almost two years until the recovered proportion drops permanently below 3%. For the reference scenario, without vaccination (dark blue line), the recovered proportion converges to an alarming 4.5% of the total population.

Our simulations show irregular, non-deterministic dynamics in the early part of the vaccination period (t ≈ 500 − 1000), a characteristic that has been so far confirmed during the 2020-2021 COVID-19 pandemic. In this early postvaccination period, we observe spikes in infection ratio which tend to disappear towards the moderate-term period. To better quantify the distinct dynamics of these two periods, and also relate them to the vaccination rate r v , we provide Figure 5 . As such, we measure the maximum (Max) value of the infected and recovered waves, as well as the final (End) value of convergence (i.e., average value of infected and recovered during the period t = 1500 − 2000). Figure 5a suggests that, as the vaccination rate r v increases, the difference between the maximum and end values of infected increases from about ×2 to ×4. Also, we notice that starting with r v ≥ 0.2 the amplitude of the maximum infection wave (orange line) remains the same. Nevertheless, the end value of the infection proportion (blue line) decreases steadily as r v increases, up to r v ≈ 0.6. Since we address a long term epidemic, we should strive for a stronger vaccination policy. Figure 5b suggests that the proportion of recovered individuals is reduced more significantly starting from r v ≥ 0.5. Thus, the difference between the maximum recovered and end value of recovered proportions increases to ×1.8 when a stronger vaccination is adopted. Finally, we depict in Figure 5c the daily number of cases, as influenced by the vaccination rate r v . We notice that the number of cases drops with r v . This rate is non-linear, but may be fitted by a logarithmic downward trend. Based on the strength of the vaccination strategy, we may reduce the daily number of cases by ×1.5 up to ×2.5, which translates into (partial to total) dampening the effect of COVID-19 waves.

All the results depicted in Figures 3-5 are based on 100 repeated computer simulations. We did not include additional error bars or confidence intervals in order not to over-complicate the visualizations, and because the convergence results (t > 1000) are highly stable in all presented scenarios.

Current scientific efforts to predict and control the COVID-19 pandemic represent global-scale efforts and include the customization of available SIR/SEIR models [15, 18, 30, 22] . However, the predominant usage of underlying homogeneous mixing models, which adopt a single population scale [6, 27, 5, 4, 12] , impede the reproduction of meaningful characteristics of real-world epidemic outbreaks. Conversely, Network Science inspired approaches can contribute to better modeling of complex systems in general [23, 31, 32] , by considering network topology, community structure, and human mobility patterns [24, 25, 26] .

This paper proposes an original intelligent information system which provides a qualitative assessment of the performance of vaccination strategies, in the COVID-19 context, summarizing an overview over the moderate-term period 2021-2025. To replicate real-world conditions, we adopt a heterogeneous population model with mobility, based on creating stochastic blocks, according to the size and geographical location of each settlement inside a chosen country [28] . Also, we propose a SIR-V epidemic model, which, unlike the classic SIR, considers patient relapse from R → S based on a relapse timeout δ r in accordance to recent epidemiological studies [34, 14, 37] . Also, SIR-V implements individual vaccination with a variable rate r v . Nevertheless, in accordance to recent studies [2] , vaccination does not offer long term immunity, rather it reduces the probability of becoming infected and the infectious rate in case of infection. This new rate λ v is updated for all vaccinated individuals. Finally, we only start the vaccination strategy after a timeout of t = 365 iterations (1 year), in compliance with the real-world situation.

We employ computer simulations of epidemic outbreaks over a ≈ 6-year period (t = 2000 iterations/days) at country level for GPM models of Romania, Germany and South Korea. The experiments reveal a high variability in the early stages of the simulation (t = 0 − 400), that suggest a correlation with the population density. Namely, denser countries have a higher chance of developing larger initial waves in case no strict isolation is imposed. After the first simulation year, and after vaccination starts (t > 400), the proportion of infected individuals has a declining convergent trend, regardless of country density. By testing different vaccination rates r v = 0 − 1 we find that during the early vaccination period (roughly 2021-2022), waves of infection are still probable for moderate vaccination strategies (r v ≤ 0.3), while stronger strategies manage to dampen subsequent waves better. During the moderate-term period (roughly 2023-2025), most vaccination strategies (r v ≥ 0.2) converge similarly, and keep the infected ratio much smaller than without any vaccination.

In favor of adopting vaccination strategies worldwide, our results suggest that, without a vaccination policy (r v = 0), the infected count would maintain a high, pseudo-constant proportion of 0.4-0.5%, that is comparable to a permanent infection wave. Weaker vaccination, at a rate of just r v = 0.2 − 0.4 (i.e., 20-40% of the total population, over 5 years), will still converge slowly towards the dynamical characteristics of the other stronger strategies. All vaccination strategies with r v ≥ 0.4 produce similar infection dynamics by maintaining the infected proportion below 0.2%, compared to 0.4-0.5%, in case of no vaccination. We observe that the pandemic is not eradicated over the 5-year simulation window, but remains as a permanent viral outbreak without significant waves (i.e., in case we implement vaccination). We believe this happens because the current vaccination is not able to offer total immunity, because of the inevitable patient relapse, and because the global vaccination is a very slow process. Finally, we quantify the reduction in daily number of cases to drop by ×1.5 up to ×2.5 for vaccination rates r v = 0.2 − 1. In general, the results offered by our realistic epidemic model encourage stronger (i.e., faster, at r v ≥ 0.4) vaccination strategies to be adopted worldwide.

Indeed, our study is still limited by our full understanding of the COVID-19 pandemic, on the performance of available vaccines, on the evolving new viral strains, and on in-depth global contact tracing data. As a future direction, we consider the update of the SIR-V model with new epidemiological findings, addition of an E (exposed) state, and heterogeneous (regional) vaccination policies. Also, real and simulated epidemic dynamics, based on time series data, could be further compared using a multi-variable similarity metric such as the statistical fidelity [29] . Overall, we believe that our study, based on the synergy of the original GPM and SIR-V models, represents a timely contribution for tackling the current COVID-19 pandemic and supporting global decision makers.

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The authors declare no competing interests.