Dynamical Regulations On Mobility and Vaccinations To Control Covid19 Spread


 Using a system of time-dynamical equations, we investigate how daily mobility indices, such as homestay above the pre-Covid normal (H%; or H-forcing), and cumulative vaccinations (Vc%; or V-forcing) impact the net reproductive rate (R0) of COVID19 in ten island nations as a prototype, and then, extending it to 124 countries Worldwide. Our H- and V-forcing model of R0 could explain the new trends in 106 countries. The disease transmission can be controlled by forcing down R0(H, Vc) < 1 with enforcement of continuous H > 40% in 91% of countries with 0% vaccinated plus recovered, Vp. The required critical H% decreases with increasing Vp%, dropping it down to 20% with 25% Vp, and further down to 8% with 50% Vp. However, the regulations on H% are context-dependent and country-specific. Our Model is useful in forecasting and controlling the disease spread when the effectiveness of the vaccines is a concern due to new variants, and/or delays in vaccination rollout programs.


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When a novel virus emerges, the community mitigation strategies, especially those concerning 27 population mobility, are the most readily available intervention to slow down the transmission 1 .

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Most countries implemented strict population mobility policies to suppress transmission of 29 SARS-CoV-2 (Covid19), and in some countries, a convincing reduction in case incidence was 30 observed, at least temporarily 2,3 . Even though many vaccination programs are rolled out World-31 wide, the current rate of the spread of Covid19 as a new infection wave with Delta and other 32 variants concerns whether the vaccines may not be effective against the new variants. The 33 reproductive rate, R0, of the Delta variant is far higher (5.08) compared to the ancestral virus 34 (2.79) 4,5 . Due to the high R0 associated with higher transmissibility, low vaccine coverage rates, 35 and lower vaccine effectiveness, the social measures will need to be strengthened to combat the 36 ever emerging variants. A high R0 also means much higher vaccine coverage rates need to be 37 achieved compared to the ancestral variant 5 . Google logins by people and location identifiers and computed as proxies for people's spatial 45 density movement as a percentage change from the pre-Covid scenarios. Nouvellet et al. 9 have 46 shown that a drop in R0 below the critical R0 = 1 requiring for disease extinction, correlates 47 with the homestay H%. Further studies also show similar findings 6,7,8 . However, the com-48 bined effect of people's mobility restraints with nations' vaccination percentages has not yet 49 been understood enough through dynamical process modeling on Covid19 disease transmission.

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Here we designed a simple study to predict the degree of population mobility restrictions 52 needed to bring the R0 below one, for countries with varying levels of cumulative vaccinations. 53 We used a system of time-dynamical equations incorporating the effect of homestay percent-54 age (H%) and the cumulative vaccinations (V c %), fitting to the data of new cases and deaths 55 obtained from the University of Oxford Covid19 database 11 , to compute the net reproductive 56 rate R0 in 124 nations. We used data from ten island nations as a prototype to select the best  Among the three alternative models fitted to the data of New cases and Deaths from ten 66 island nations as a prototype, Model 3 (M HV ) that combined the effects of both the homestay 67 (H%) and the vaccination (V c %), fitted the best, in general, as per the Akaike information 68 criterion (AIC) ( We extended the M HV model-fitting and the analyses to the World data of 124 countries 78 (see the Appendix). The model M HV could explain the variations and the trends in the data 79 in 106 countries well w.r.t. the trends in their respective residuals. Ninety five out of the 106 80 countries showed a calibratable functional relationship between the R0 and the H% (Fig. 1), 81 given their respective percentages of the people vaccinated plus the number recovered, denoted 82 by V p %. Note that V p % >= the percentage vaccinated, V c %. The eleven out of the 106 coun-83 tries that did not show a marked variation in the R0 with respect to an increase in the H% 84 was because the variation in the H% data was not sufficient to capture such changes in the 85 functional response of R0. That is, for those nations, the functional response of R0 with respect 86 to H% was non-calibratable, other than the values of R0 for their given V p %.

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When the H% is increased, the R0 decreased for nations at different percentages of vacci-89 nations plus recovered, V p : 0%, 25%, and 50%, in the said 106 countries (Fig. 1). The model 90 M H V indicated that 91% of the nations, with both mobility regulations and effective vacci-91 nation programs, out of the 106, requires a minimum of 40% homestay above the pre-Covid 92 normal to bring the R0 < 1 from the status quo ( Fig. 1). This threshold of H% at R0 = 1 93 lowers with the increasing percentage of the cumulative vaccinations. For example, with > 25% 94 more vaccinated (V c %) plus the recovered, the H% at R0 = 1 lowers down to 20% in 97% of 95 the 106 countries. For countries with > 50%V p , the H% requiring for R0 < 1 turns out to 96 be about 8 for about 99% of the 106 nations (Fig. 1). Figure. 2. shows how H% at R = 1 97 decreases with the increasing V p %, vaccination plus recovered percentage. Overall, the R0, 98 averaged over the last seven days, is negatively correlated with the percentage-vaccinated in 99 the nations, with R 2 = 0.41 and p < 0.01 (Fig. 65 in the Appendix). It indicates that > 80% 100 vaccinated in a nation has not fully guaranteed a R0 < 1, with upper 95% confidence interval 101 4 crossing R0 = 1 at 70% vaccinated. The detailed countrywide plots are given in the Appendix. The four nations presented in Figure 3 and 4, among the ten island nations in the proto-106 type analysis: Australia, Taiwan, Sri Lanka, and the UK represent contrasting levels of H% vs 107 V c %. Three of them, namely, Taiwan, Sri Lanka, and the UK showed a significant reduction 108 in R0, but not Australia. In Taiwan, the decline was due to an increased mobility restriction 109 forcing a hike in H%, but not due to vaccinations, while in the UK, it was due to escalating  The simulation results of island nations in Figure 4 show the level of H% needed for R0 < 1, 124 indicating that the disease can be forced to go extinct, i.e., R0(H) < 1, from the status quo 125 (i.e., with the already administered V c % of the respective countries plus recovered), with an 126 enforcement of a continuous homestay of H% > 20, from the pre-Covid normal. This is the 127 same case for many other in the 106 nations given in the Appendix. The R0 vs. H% panels 128 in Figure 4 show that the percentage of vaccination, V c %, brings the required critical H% to  indices will be country-specific, and will need ground-truthing. It will be more informative to 154 model the same in spatially explicitly in countries such as Japan and other, where the regional 155 6 data on new cases, deaths, and mobility are available.    Firstly, we took ten island nations greater than 25000 to 8.0E6 km 2 in land area, ranging 194 from Haiti to Australia, with countries' populations ranging from 10 to 270 million, to test 195 our prototype disease dynamical models that we developed. We tested three alternative nested  to be a conservative 5 to 13 days backwards in time from day t, of the individuals belong in the 215 above integral that are infectious as at day t. We assume that the recovered individuals never 216 get re-infected during the modeling time, which is 144 days since the beginning of the new Covid 217 wave. Thus, at the beginning of the process, we write for the non-vaccinated scenario, where D(t) are the number of daily deaths.  We consider the observed, or the identified new cases, C(t), are immediately self-quarantined 233 or hospitalized in the case of Covid19, such that, they are isolated from being able to infect 234 9 other individuals in the community, that is, being removed from the I r (t), not being considered 235 for the mass-action effect in contributing to further disease spread. We take the probability 236 of identification of the infected from the currently infectious as ǫ, allowing its range to be be-237 tween a conservative, 0.15 to 0.6, in the parameter estimations. It is known that symptomatic 238 percentage was 13 − 18% 15 in Covid19. Thus, we further allowed a provision for being some 239 identified in random checking. also has already been taken off from the I(t) similar to C(t), out of the mass-action effect.

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Thus, a discrete time-dynamical community Covid19 infection model can be written as (1) where, ∆t = 1 is one day in our study. whereas, when it is < 1, the number infected in the community declines. Incorporating the H-forcing, s.t., β = γ(1 − θ(H(t)/100) k ), thus, we can write, This is a metric of net reproductive rate R0 of the infection written as function of H-and 269 V-forcing over time.    (c) V p = 50% Figure 1: The World data: The net reproductive rate R0 vs. the percentages of Home-stay H% at different percentages of the population vaccinated plus recovered, V p % The R0 decreases with the increasing H% at (a) V p = 0%, that is 0% is vaccinated plus recovered from the susceptible, (b) V p = 25%, and (c) V p = 50%. Here, we plotted the 106 out of the 124 nations based on the estimated M HV model that explained the variation in the nations' respective data. The 95 nations out of the 106 allowed enough variation in the degree of H% above the pre-Covid normal to make it possible to calibrate the R0 vs. H% functional relationship based on the model. (Note that V p >= V c , where V c is the vaccinated population percentage). The functional relationship: R0 = γΨ(1 − θ(H/100) k ) − ǫ, where Ψ is the susceptible population proportion, that is, the proportion of the total population N minus the effective number out of the vaccinated, νV c ,minus the number recovered, assuming ν as the likelihood that a vaccinated individuals is not re-infected, or as a proxy for the average efficacy of the vaccines. The R0 < 1 indicates the threshold below which there is a tendency for the disease to go extinct. (see Appendix for country-specific graphs).  Table.1 and  also the Table.1 in the Appendix. The model fitted to 124 countries are given in the Appendix.    The projected percentages of homestay H% and vaccinations plus recovered, V p %, required to bring R0 below 1, for the ten island nations (the prototype) The Akaike Information Criteria (AIC) values of the fitted alternative M H , M V and M HV models, and the critical values of H(R0 = 1)% and V p (R0 = 1)% computed based on the all-representative M HV model are given (see graphs in the Appendix for all nations). The coefficient ν, a proxy for the efficacy of the vaccines, was set at 0.8 in island estimations. Thus, the degrees of freedom (df) in both M H and M HV models become 4. NLL-Negative log likelihood of the model fits. And µ is the daily death rate. Other parameter values are given in the Table.1 in the Appendix. The graphs of H(R0 = 1)% with respect to V p %, and R0 with respect to H% based on the M HV are given in the Appendix. Note: ap stands for, as at present.