key: cord-0980946-fsom1eb4 authors: Tetteh, Josephine N.A.; Kinh Nguyen, Van; Hernandez-Vargas, Esteban A. title: Network Models to Evaluate Vaccine Strategies towards Herd Immunity in COVID-19 date: 2021-09-09 journal: J Theor Biol DOI: 10.1016/j.jtbi.2021.110894 sha: bde520457505920f6d2b6e11999fb9600040648c doc_id: 980946 cord_uid: fsom1eb4 Vaccination remains a critical element in the eventual solution to COVID-19 public health crisis. Many vaccines are already being mass produced and supplied to the population. However, the COVID-19 vaccination programme will be the biggest in history. Reaching herd immunity will require an unprecedented mass immunisation campaign that will take several months and millions of dollars. Using different network models, COVID-19 pandemic dynamics of different countries can be recapitulated such as in Italy. Stochastic computational simulations highlight that peak epidemic sizes in a population strongly depends on the social network structure. Assuming a vaccine efficacy of at least 80% in a mass vaccination program, at least 70%of a given pop ulation should be vaccinated to obtain herd immunity, independently of the social structure. If the vaccine efficacy reports lower levels of efficacy in practice, then the coverage of vaccine tion would be needed to be even higher. Simulations suggest that the “Ring of Vaccination” strategy, vaccinating susceptible contact and contact of contacts, would prevent new waves of COVID -19 meanwhile a high percent of the population is vaccinated. The novel Coronavirus SARS-CoV-2 epidemic which first emerged from Wuhan, China in 27 December 2019 has spread globally, causing high levels of mortality and morbidity worldwide. 28 To curb the spread, governments across the world have implemented measures ranging from 29 quarantining, social distancing, wearing of face masks among others. Amidst this crisis, 30 national health care systems such as in Italy and the United States of America have been 31 overwhelmed by the ever-increasing number of infection cases [56] . 32 In studying infectious diseases, mathematical models play a significant role in estimating 33 disease transmission parameters as well as the severity and intensity of an outbreak [12] . The The assumption of random homogeneous mixing in epidemiological models has been doc-number of connections a node has to other nodes). In small-world networks such as the Watts 66 and Strogatz model, the pattern of connectivity between nodes is more localized [82] and the 67 average path length is comparable with a homogeneous random network, without any regard 68 to clustering. 69 During a disease outbreak, it is less likely for the disease to reach epidemic proportions 70 in the power-law network than it is in random networks [23, 81] . This is because power-law A key question to be answered is, how much vaccine is required to create herd immunity to 91 block SARS-CoV-2 transmission? [5] . In other words, how many people need to be vaccinated 92 in order to reach herd immunity? Here, we employ a network-based approach to explore 93 the potentials of two vaccination schemes, classical mass vaccination and ring vaccination, in minimizing the spread of SARS-CoV-2, see Figure 1 . Ring vaccination is a vaccination 95 strategy in which infected cases and contacts of cases are identified and vaccinated [32, 60] . 96 This strategy is especially efficient in controlling rare pathogens and has been successful in probability of infection, β(t) is as follows: where b 1 is the first boundary (i.e. function value at time zero), b 2 is the second boundary, r 1 158 is the rate of change of first period, r 2 is rate of change of second period, m 1 is the midpoint 159 of the first period (start of interventions), m 2 is the midpoint of the second period (end 160 of interventions) and t is time. Note that if b 1 > b 2 the function increases first and then 161 decreases, and vice versa. To get a good set of parameters for β(t), we applied Eqn (1) to 162 emulate infection data (from February 22 to September 1 2020) from Italy, one of the worst 163 hit countries during the SARS-CoV-2 pandemic (see Figure 2 ). The main goal of vaccination in this model is to prevent transmission. At the beginning 165 of each epidemic simulation, a fraction of the population (%vac) is given a vaccine. move to V state. Individuals in V that become exposed to the virus due to contact with an For the first strategy which we refer to as the classical mass vaccination strategy, a frac-177 tion of individuals start in the V state (denoted %vac) with one index case (patient zero) and 178 the rest in the S state. The population coverage in this case ranges from 10%, 20%, · · · , 100%. The modelling process follows as in Algorithm 1. For the second strategy which we refer to as ring vaccination, vaccination occurs after 181 a percentage (1% or 3%) of the population has been exposed to the virus during the epi-182 demic. In this case, we simulate the epidemics as described above (in Algorithm 1) with 183 no vaccination and only begin vaccine administration after a proportion of the population 184 (%exposed) has been exposed (See Algorithm 2 for more details). We assume that once a tracing. Contacts of contacts are also vaccinated with the same probability. In our model, 190 we also assume that traced and identified susceptible contacts and contacts of contacts are 191 vaccinated. This process is described in Algorithm 2 and detailed implementation can be 192 found at https://github.com/systemsmedicine/COVID-19-Network-Model. In both strategies, we assume that the vaccine does not have an effect on infectious or 194 asymptomatic individuals. Table 1 summarizes all parameters and key terms used in this 195 study. The model (depicted in Algorithm 1 proceeds in discrete one-day time steps for a period 199 of 360 days to determine disease dynamics. Each node i = 1, · · · , N has an individual state x it at time t. We initialize the model 201 simulation by randomly assigning #I ∈ (0, N ) nodes to the infectious state (I), that is 202 setting x i0 = I, i ∈ I 0 and the rest of the nodes to the susceptible state, x j0 = S for j ∈ S 0 . 203 We denote by X t the set of nodes with state x it = X at time t. Conditional on current state 204 x it , the next state x it ′ for node i is determined as follows. • Susceptible nodes • Asymptomatic nodes • Infectious nodes • Vaccinated nodes S within 14 days post vaccination V after 14 days post vaccination V otherwise 10% − 100%. For ring vaccination, the proportion of exposed individuals (%exposed) varied 213 as 1% and 3% (see Table 2 ). For ER simulations: N = 10 6 , b 1 = 0.028, b 2 = 0.001, r 1 = 0.09, Implementations of this network model was computationally demanding and challenging using 219 conventional resources. For instance, one simulation can take up to several hours or days 220 to complete in a modern desktop computer. Thus, due to these limitations, we employed a In a completely susceptible population, the introduction of one exposed individual leads to 236 the spread of the infection with more than one peak of cases of infection after some months. In Figure 3 , the mean number of infectious cases peaks at 0.83% for ER network and 0.07% 238 for BA network. These scenarios determine the outcome of vaccination at the initial stages of the epidemic 241 before infectious cases peak in both networks. From Figure 4 and Tables S1 and S4, we 242 observe that generally, infection peaks are much lower in Barabasi-Albert network than in the 243 Erdos-Renyi network. In the Erdos-Renyi network, when vaccine efficacy is 40%, a population coverage of 40% or 245 more is needed to achieve infection peak with less than 1% infection cases in the population. Table S1 (and also Figure 4 ) reveals that with vaccine efficacy of 60%, a coverage more than 247 60% keeps mean infection cases on the low with elimination of the peak occurring when 248 coverage is more than 80%. Furthermore, low cases of infection are observed when 70% 249 or more of the population is vaccinated and vaccine efficacy is 80%. On the other hand, a 250 vaccine which is 100% efficacious requires just 20% or more of the population to be vaccinated 251 to ensure there is no peak of infections. In the Barabasi-Albert network, with a vaccine efficacy of 40%, a population coverage 253 of more than 60% ensures elimination of infection peaks whereas when vaccine efficacy is 254 60%, a coverage more than 70% achieves elimination. In addition, when 70% or more of the 255 population is vaccinated with an 80% efficacious vaccine, infection cases are almost negligible. In the case when a 100% efficacious vaccine is administered, a population coverage more than 257 20% keeps infection peaks at bay. (a) Erdos-Renyi Model: Heatmap illustrating average infection peaks considering vaccine efficacy and population coverage. Derived from Table S1 . (see Figures 7a, 8a, 7b, 8b and Tables S6, S5, S3, 268 S2 ). In both networks, a 100% efficacious vaccine ensures that no susceptible (unvaccinated) 269 individual gets exposed to the disease (Figure 7) . Similar results are observed in the proportion 270 of vaccinated individuals who later become exposed (Figure 8) 282 We carried out simulations using Algorithm 2 to determine the infection outcome when ring 283 vaccination is used in both networks. In Figures 9 and 10 (see also Figures S5, S6 , S11, S12 284 and Table 3 ), we show these outcomes with varying scenarios of vaccine efficacy and when 1% 285 or 3% of the population is already exposed to the disease before the onset of vaccination. 286 We see that in both networks, there are less cases realised with a 1% exposed population 287 before the start of vaccination as compared to a 3% exposed population. In addition, the 288 number of infected cases in these scenarios are considerably lower than that of the classical 289 vaccination method. Also, even with a 100% efficacious vaccine, total eradication of the peak 290 is not achieved irrespective of the exposed population prior to vaccination and regardless of 291 the network. This can be seen in Figure S5 , S6, S11 and S12. 292 Table 3 : Average population coverage (in %) for ring vaccination scenarios in both ER and BA networks considering vaccine efficacy. Vaccine Efficacy 40% 60% 80% 100% (ER Network) (1% of population exposed before vaccination) 0.71 ± 0.20 0.77 ± 0.23 0.67 ± 0.20 0.68 ± 0.19 (3% of population exposed before vaccination) 0.81 ± 0.23 0.89 ± 0.26 1.00 ± 0.29 0.65 ± 0.19 (BA Network) (1% of population exposed before vaccination) 0.062 ± 0.02 0.068 ± 0.02 0.043 ± 0.01 0.032 ± 0.01 (3% of population exposed before vaccination) 0.073 ± 0.02 0.049 ± 0.02 0.057 ± 0.02 0.65 ± 0.19 In comparison to mass vaccination, a lower percentage of the population has to be vacci-293 nated when using a ring vaccination protocol in order to attain low infection cases (see Table 294 3). This is especially so as with effective contract tracing, more individuals can be vaccinated 295 and thus decreasing the number of infections. It is worth noting also that even with the 296 above results, the percentage of vaccinated individuals is lesser when only 1% of the popu-297 lation is exposed prior to vaccination than when the prior exposed population is 3%. Also, from Table 3 we see that for each network, the vaccinated populations are very similar in 299 both scenarios (that is when 1% or 3% of the population is exposed) respectively irrespective 300 of the efficacy of the vaccine. (a) (b) Figure 9 : Outcome for ring vaccination scenarios for each vaccine efficacy percentage on a Erdos-Renyi network. This shows the changes in mean infected cases over time when there is 1% and 3% prior exposed population in (a) and (b) respectively. the BA network than in the ER network. In addition, more cases are seen when 3% of the 305 population is exposed than when 1% is exposed. From our simulations for ring vaccination in 306 both networks, vaccinated individuals remain vaccinated and do not get exposed during the 307 course of the infection process. 308 Table 4 : Average proportion (in %) of unvaccinated (susceptible) individuals who got exposed in the course of the infection in both ER and BA networks under ring vaccination. Vaccine Efficacy 40% 60% 80% 100% (ER Network) (1% of population exposed before vaccination) 0.71 ± 1.00 0.82 ± 1.01 0.77 ± 1.04 0.71 ± 0.99 (3% of population exposed before vaccination) In this study, we modelled the spread of SARS-CoV-2 infection using an SAIRV model 330 structure on social networks. We run stochastic simulations to determine vaccine efficacy and 331 population coverage limits capable of extinguishing the disease. We considered two vaccination 332 strategies, each with varying scenarios regarding vaccine efficacy and population coverage. 333 We found that, the introduction of a single infectious person into a completely susceptible 334 population leads to the spread of infection giving rise to more infectious cases and subsequently 335 more than one infection peak. This is an indication that in the absence of a good enough 336 The model also assumes that all infectious people recover from the disease and are immune. The effect of mortality on the dynamics of the disease is not considered as this model aims 395 to study the general transmission dynamics and the effects of varying vaccination strategies. [68] Pfizer Inc. Pfizer and biontech announce vaccine candidate against covid-598 19 achieved success in first interim analysis from phase 3 study, 2020. URL https://www.pfizer.com/news/press-release/press-release-detail/ 600 pfizer-and-biontech-announce-vaccine-candidate-against. Accessed: 2020-12- The Lancet Public Health, 5(5):e261-e270, 2020. [71] C. L. Ricardo-Azanza and E. A. Vargas-Hernandez. Figure S5 : Erdos-Renyi network: Distribution of final infectious cases in different timing for ring vaccination scenario when 1% of the population is exposed prior to vaccination. Circles represent mean infection cases for each month connected by lines. Figure S6 : Erdos-Renyi network: Distribution of final infectious cases in different timing for ring vaccination scenario when 3% of the population is exposed prior to vaccination. Circles represent mean infection cases for each month connected by lines. 0.1 ± 0.03 0.1 ± 0.03 0.14 ± 0.04 0.05 ± 0.01 20% 0.21 ± 0.07 0.13 ± 0.05 0.10 ± 0.04 0.01 ± 0 30% 0.07 ± 0.03 0.08 ± 0.03 0.09 ± 0.03 0 ± 0 40% 0.08 ± 0.03 0.08 ± 0.03 0.04 ± 0.02 0 ± 0 50% 0.04 ± 0.01 0.09 ± 0.03 0.04 ± 0.02 0.0 ± 0 60% 0.04 ± 0.01 0.04 ± 0.01 0.01 ± 0 2.84 ± 16.12 7.39 ± 23.53 1.56 ± 6.39 0.0 ± 0 60% 3.9 ± 19.1 3.52 ± 17.24 0.18 ± 1.81 0.0 ± 0 70% 0.0± 1.87 ± 13.07 0 ± 0 0 ± 0 80% 0 ± 0 0 ± 0 0 ± 0 0 ± 0 90% 0 ± 0 0 ± 0 0 ± 0 0 ± 0 100% 0 ± 0 0 ± 0 0 ± 0 0 ± 0 Figure S11: Barabasi-Albert network: Distribution of final infectious cases in different timing for ring vaccination scenario when 1% of the population is exposed prior to vaccination. Circles represent mean infection cases for each month connected by lines. Figure S12: Barabasi-Albert network: Distribution of final infectious cases in different timing for ring vaccination scenario when 3% of the population is exposed prior to vaccination. Circles represent mean infection cases for each month connected by lines. Collective dynamics of small-worldnetworks. nature Modeling shield immunity to reduce COVID-19 epidemic spread Harnessing case isolation and 647 ring vaccination to control ebola Estimation of the time-varying reproduction number of covid-19 outbreak 654 in china Simulation of the covid-19 pandemic on the 656 social network of slovenia: estimating the intrinsic forecast uncertainty