Vaccination strategies in structured populations under partial immunity and reinfection

  1. Gabriel Rodriguez-Maroto
  2. Iker Atienza-Diez
  3. Saúl Ares
  4. Susanna Manrubia

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Abstract

Optimal protocols of vaccine administration to minimize the effects of infectious diseases depend on a number of variables that admit different degrees of control. Examples include the characteristics of the disease and how it impacts on different groups of individuals as a function of sex, age or socioeconomic status, its transmission mode, or the demographic structure of the affected population. Here we introduce a compartmental model of infection propagation with vaccination and reinfection and analyse the effect that variations on the rates of these two processes have on the progression of the disease and on the number of fatalities. The population is split into two groups to highlight the overall effects on disease caused by different relationships between vaccine administration and various demographic structures. We show that optimal administration protocols depend on the vaccination rate, a variable severely conditioned by vaccine supply and acceptance. As a practical example, we study COVID-19 dynamics in various countries using real demographic data. The model can be easily applied to any other disease and demographic structure through a suitable estimation of parameter values. Simulations of the general model can be carried out at this interactive webpage [1].

Author summary

Vaccination campaigns can have varying degrees of success in minimizing the effects of an infectious disease. It is often very difficult to assess a priori the importance and effect of different relevant factors. To gain insight into this problem, we present a model of infection propagation with vaccination and use it to study the effects of vaccination rate and population structure. We find that when the disease affects in different ways distinct population groups, the best vaccination strategy depends non-trivially on the rate at which vaccines can be administered. The application of our analysis to COVID-19 reveals that, in countries with aged populations, the best strategy is always to vaccinate first the elderly, while for youthful populations maximizing vaccination rate regardless of other considerations may save more lives.

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    Reply to the reviewers

    Reviewer #1 (Evidence, reproducibility and clarity (Required)):

    The paper tackles an important problem regarding the effect of demographic dependent vaccination protocols on the reduction in the number of deaths with respect to the situation of no vaccination (say J). A compartmental SIRD model with reinfection Y is proposed, stratified in two (age dependent) groups, based on a binary reduction of a given contact map, and given infection fatality risk (IFR). Several countries are then analyzed.

    As far as I understand we have a control variable v, parameters of the stratified model (i=1,2) tuned to match IFRi, and a control objective, i.e. minimization of J over one year.

    The paper is well written. The final message and some theoretical passages are not completely clear, at least to me. I have the following observations that the authors may want to consider.

    We thank the referee for the revision and are very glad that the overall evaluation is positive. Comments and suggestions have been thoroughly addressed, as we discuss in the following.

    1) The study of stability of infection free and endemic equilibria should be better developed. The 5 equations can be reduced to 4 (neglecting D) and the characteristic of the reduced Jacobian used to characterize the local asymptotic stability of equilibria, instability, bifurcation points etc... Alternatively, one can use a co-positive Lyapunov function (LF). For instance, if we take the LF V=S+I+Y+R, we get $\dot V=-\mu_I I-\mu_Y Y \le 0$. If $\mu_I$ and $\mu_y$ are strictly positive all equilibria are characterized by (S*,0 0,R*) and D=1-S*-R*. So, I don't understand the phrase after (7,8), notice that Y cannot be zero in finite time. For $\mu_y=0$ then Y* can be nonzero. I guess that closed-form computation of S* and R* is possible as function of the parameters at least in the case v=0. The stability result should be cast in function of the current reproduction number (not explicitated) wrt to S and R.

    The authors are invited to have a look at

    1.1) Pagliara et al, "Bistability and Resurgent Epidemics in Reinfection Models", IEEE CSLetters, 2018,

    for a theoretical analysis of stability on a similar (just a little bit simpler) model.

    We appreciate the suggestions of the referee for improvement of this material. We have carried out an in-depth revision of the stability analysis and significantly extended it. The major addition has been, as suggested, a section relating the current reproductive number at equilibrium (we call it the asymptotic reproductive number in the text) to the fixed points of the dynamics for three different scenarios: general model, no vaccination, and zero mortality of reinfected individuals. As Pagliara et al. show in their paper, the connection between the fixed points and the reproductive number is not trivial, but it is possible to derive it through the next-generation matrix technique, as we now do. Additional references regarding this technique have been added. We have included a Table summarizing the stability analysis (page 2 in SI 3) at the end of this new section.

    Other modifications include the reduction of 5 equations to 4 for the stability analysis and a clarification of possible equilibria (page 1 of SI 3), rephrasing and correcting our sentence after eqs. (7) and (8). We also attempted to obtain a closed-form computation of S* and R* but, to the best of our knowledge, concluded that it is not possible. We would be happy to pursue any insight in this respect the referee may have.

    What said before should be also extended to the stratified model, where a "network" Rt could be defined, see for instance

    1.2) L. Stella et al, "The Role of Asymptomatic Infections in the COVID-19 Epidemic via Complex Networks and Stability Analysis", SIAM J Cont. Opt., 2021, (arxiv.org/pdf/2009.03649.pdf)

    We thank the referee for pointing out this reference. Following the analysis in Stella et al., we have carried out a stability analysis for the stratified model as well. The results are included in a new section (pages 7-10 in the SI 3).

    2) It is not clear whether the free contagion parameters of the model have been fitted on real data (identification from infection and reinfection data). Notice that the interplay between vaccination strategies and NPI is important, see e.g.

    *2.1) Giordano et al, Modeling vaccination rollouts, SARS-CoV-2 variants and the requirement for non-pharmaceutical interventions in Italy", Nature Medicine 2021, *

    where progressive vaccination in reverse age order is considered together with different enforced NPI countermeasures.

    In the first part of our study, parameters are intendedly left free because we aim at describing the generic behavior of the model. Still, we derive several inequalities and relationships between parameter ratios that seem to be sensible attending to what the different classes in the model stand for. This is as described in sections regarding model parameters when the two generic models (SIYRD and S2IYRD) are introduced. The aim is to represent both the generic dependence with some variables and a broad class of contagious diseases, so parameters are mostly free. In agreement with this approach, parameters can be also freely varied in the companion webpage.

    In the second part of our study, the model is applied to COVID-19. In that case, we have used parameter values in agreement with observations, as (admittedly poorly) explained in pages 9-10 of the main text. Indeed, not enough information on parameter estimation was provided in the main text, and the SI 2 also needed some additional information. This has been amended. Let us explicitly mention that we have not fitted the dynamics of the model to any actual data set to fix specific values, as Giordano et al. do. In our case, we have first used different demographic data sets to evaluate contact rates and IFRs of the two population groups (these are parameters Mij and Ni in eqs. (7-10)). Secondly, recovery and death rates are estimated through the IFRi values for each age group i and the infectious period of COVID-19, that we fix at dI=13 days. Third, infection rate βSI=R0/dI has been estimated fixing R0=1, since the reproductive number of COVID-19 all over the world fluctuates around this value (Arroyo-Marioli et al. (2020) Tracking R of COVID-19: A new real-time estimation using the Kalman filter, PLoS ONE 16(1):e0244474). The reinfection rate is defined through its relationship with the infection rate, βRI= α1 βSI, where α1 was in the range 0-0.011 at early COVID-19 stages (Murchu et al. (2022), Quantifying the risk of SARS‐CoV‐2 reinfection over time, Rev Med Virol 32:e2260) and seems to be about 3-4 fold larger for the omicron variant (Pulliam et al., Increased risk of SARS-CoV-2 reinfection associated with emergence of the Omicron variant in South Africa, www.medrxiv.org/content/10.1101/2021.11.11.21266068v2). Given the relationships derived among parameters, our only free parameter was α2RY= α2 βRI, and we fixed it to α2=0.5 (i.e., reinfected individuals recover twice as fast as individuals infected for the first time).

    Once more, it was not our goal to precisely recover specific trajectories of COVID-19 or to point at possible future scenarios, but to illustrate the dependence of major trends with model parameters. Also, the appearance of new variants requires the reevaluation of parameters. For example, omicron has different IFR (therefore different mortality and recovery rates), a different infectious period, and higher infection and reinfection rates. In this context, the interactive webpage (where we will update demographic profiles and IFR data as they become available) is a useful resource to simulate any situation different from current or past ones.

    3) In the model the immunity waning is not explicitly considered (flux from R to S or better from a vaccinated compartment to S). It is clear that this complicates the model. Please discuss why the indirect way the waning is considered here is justified.

    3.1) Batistela et al, "SIRSi compartmental model for COVID-19 pandemic with immunity loss", Chaos Soliton and fractals, 2021.

    3.2) McMahon et al, "Reinfection with SARS-CoV-2: Discrete SIR (Susceptible, Infected,Recovered) Modeling Using Empirical Infection Data", JMIR Public health and surveillance, 2020.

    Though the model does not consider an incoming flux of individuals to compartment S, the existence of a "backward" flux from R to Y yields a transient phenomenology analogous to models with increases in the S class. Indeed, it is these fluxes that cause persistent endemic states; otherwise, the S class is monotonously depleted until infection extinction.

    In Batistela's et al. work, the possibility that individuals become reinfected is effectively implemented through a flux between the R and S classes, since only one class of infected individuals is considered and recovered individuals cannot be infected again. In our case, feeding back to S would mean that previous immunity is completely lost or that vaccines are not effective at all for some individuals. This is neither what McMahon et al. conclude when evaluating real data nor what more recent surveys indicate (see for instance the Science Brief published in October 2021 by the CDC, SARS-CoV-2 Infection-induced and Vaccine-induced Immunity, https://www.cdc.gov/coronavirus/2019-ncov/science/science-briefs/vaccine-induced-immunity.html).

    This nonetheless, complete immunity waning (feedback to the S class) and reinfections (feedback to a partly immune class experiencing overall lower severity of the disease) are equivalent to a large extent: the trend of COVID-19 seems to indicate that our Y class will be the "new S", and that fully naive individuals would arrive mostly due to demographic dynamics (birth and death processes, as also implemented by Batistela et al.). Summarizing, complete immunity waning is rare in the time scales considered in our simulations, while partial immunity that decreases the severity of the disease (after infection or vaccination) is the rule, in agreement with our choices.

    4) Reduction of deaths wrt no vaccination is of course important, but also reduction of stress in hospitals. This is particularly important now with the advent in Europe of the omicron variant. Please discuss on the real message you want to convey to policy makers in the actual scenario of the pandemic.

    The model in this work is deliberately simple. Our main goal was to explore the qualitative effects of demographic structure and disease parameters in protocols for vaccine administration. This was the reason to consider a mean-field model in a population structured into two groups. The main conclusion is that optimal vaccination protocols are demography- and disease-dependent. If this is so in our streamlined model, the more it will be in more realistic models, where one should include a finer stratification and, in all likelihood, heterogeneity in contagions. Our main message, therefore, is that there is no unique protocol for vaccine roll-out, valid for all populations and diseases. The abstract has been modified to highlight this conclusion.

    Some qualitative considerations also allow us to draw preliminary conclusions on the reduction of stress in hospitals. Since the number of hospital admissions is proportional to the incidence of the disease, the number H of hospitalized individuals can be represented as H=a I + b Y, with a>>b due to the partial immunity of vaccinated or recovered individuals (which belong to class Y upon (secondary) contagion). Therefore, minimizing the burden on the healthcare system amounts to minimizing the number of individuals in the I class. Beyond non-pharmaceutical measures, I is minimized when individuals are transferred as fast as possible to the Y class, that is, maximizing vaccine supply and acceptance. In terms of our model parameters, this entails maximizing v and also θ (the maximum fraction of individuals eventually vaccinated), for instace through devoted awareness campaigns. These ideas have been included in the Discussion section.

    Reviewer #1 (Significance (Required)):

    The final message and some theoretical passages are not completely clear, at least to me.

    Please discuss on the real message you want to convey to policy makers in the actual scenario of the pandemic.

    As discussed above, we have modified the manuscript following the advice given by the Reviewer. We think that both the presentation and the theory are clearer now.

    Reviewer #2 (Evidence, reproducibility and clarity (Required)):

    In this paper, a compartmental model of the propagation of an infection with vaccination and reinfection is studied. The impact that changes in the rates of these two processes have on disease progression and on the number of deaths is analyzed. In order to highlight the overall effect of the demographic structure of populations and the propagation of a given disease among different groups, the population is divided into two subpopulations and the model is extended to the two-dimensional case. In addition to the study of equilibria and their relative stability, the model is then applied in the case of COVID-19. Different vaccination strategies are studied using real demographic data and with a population split between under 80 and over 80 individuals. It is observed that for low vaccination rates, the advisable strategy is to vaccinate the most vulnerable group first, in contrast to the case of sufficiently high rates, where it is appropriate to vaccinate the most connected group first. The simulations show also that with a low fatality ratio, the strategy that yields the greatest reduction in deaths is vaccination of the group with the most contacts, while the situation is reversed for higher fatality ratio.

    The model and simulations presented are interesting and valuable. The comparison of the behavior of the model in the 4 different countries is very interesting, as well as the webpage created by the authors.

    We thank the referee for the very positive evaluation and are very glad that the study is found interesting and valuable.

    As minor comment, I think that the introduction of the model needs a more extensive literature review. For example, there is no mention of the classic SIR model of Kermack and McKendrick (1927) and other works on the introduction to epidemic models, which form the basis of the model presented by the authors.

    The referee is right. There is a long history of extensions and applications since Kermack & McKendrick introduced the SIR model that we obviated. This has been amended by adding an introductory paragraph with several new references at the beginning of the Models section, page 3 in the main text.

    Reviewer #2 (Significance (Required)):

    The model presented by the authors is quite original and simple enough to be suitable to different contexts and scenarios.

    Compared to previous work, this paper makes a twofold contribution, as explained by the authors. First, the introduction of reinfections shows the existence of long transients (or quasi-endemic states) that may precede the transition to a truly endemic state predicted for COVID-19. Second, the simplicity of model allows the characterization of systematic effects due to, at least, group size, demographic composition, and IFRs.

    I am involved in the study and analysis of epidemic models accompanied by network effects. I think this paper is a good contribution, although preliminary, in the analysis of the vaccination process and in the search for the optimal strategy.

    We thank the Reviewer and are glad that our goal, offering a model as simple as possible to obtain meaningful conclusions, is appreciated.

  2. Review Commons

    Note: This preprint has been reviewed by subject experts for Review Commons. Content has not been altered except for formatting.

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    Referee #2

    Evidence, reproducibility and clarity

    In this paper, a compartmental model of the propagation of an infection with vaccination and reinfection is studied. The impact that changes in the rates of these two processes have on disease progression and on the number of deaths is analyzed. In order to highlight the overall effect of the demographic structure of populations and the propagation of a given disease among different groups, the population is divided into two subpopulations and the model is extended to the two-dimensional case. In addition to the study of equilibria and their relative stability, the model is then applied in the case of COVID-19. Different vaccination strategies are studied using real demographic data and with a population split between under 80 and over 80 individuals. It is observed that for low vaccination rates, the advisable strategy is to vaccinate the most vulnerable group first, in contrast to the case of sufficiently high rates, where it is appropriate to vaccinate the most connected group first. The simulations show also that with a low fatality ratio, the strategy that yields the greatest reduction in deaths is vaccination of the group with the most contacts, while the situation is reversed for higher fatality ratio.

    The model and simulations presented are interesting and valuable. The comparison of the behavior of the model in the 4 different countries is very interesting, as well as the webpage created by the authors.

    As minor comment, I think that the introduction of the model needs a more extensive literature review. For example, there is no mention of the classic SIR model of Kermack and McKendrick (1927) and other works on the introduction to epidemic models, which form the basis of the model presented by the authors.

    Significance

    The model presented by the authors is quite original and simple enough to be suitable to different contexts and scenarios.

    Compared to previous work, this paper makes a twofold contribution, as explained by the authors. First, the introduction of reinfections shows the existence of long transients (or quasi-endemic states) that may precede the transition to a truly endemic state predicted for COVID-19. Second, the simplicity of model allows the characterization of systematic effects due to, at least, group size, demographic composition, and IFRs.

    I am involved in the study and analysis of epidemic models accompanied by network effects. I think this paper is a good contribution, although preliminary, in the analysis of the vaccination process and in the search for the optimal strategy.

  3. Review Commons

    Note: This preprint has been reviewed by subject experts for Review Commons. Content has not been altered except for formatting.

    Learn more at Review Commons

    Referee #1

    Evidence, reproducibility and clarity

    The paper tackles an important problem regarding the effect of demographic dependent vaccination protocols on the reduction in the number of deaths with respect to the situation of no vaccination (say J). A compartmental SIRD model with reinfection Y is proposed, stratified in two (age dependent) groups, based on a binary reduction of a given contact map, and given infection fatality risk (IFR). Several countries are then analized.

    As far as I understand we have a control variable v, parameters of the stratified model (i=1,2) tuned to match IFRi, and a control objective, i.e. minimization of J over one year.

    The paper is well written. The final message and some theoretical passages are not completely clear, at least to me. I have the following observations that the authors may want to consider.

    1)The study of stability of infection free and endemic equlibria should be better developed. The 5 equations can be reduced to 4 (neglecting D) and the characteristic of the reduced Jacobian used to characterize the local asymptotic stability of equlibria, instability, biforcation points etc... Alternatively, one can use a co-positive Lyapunov function (LF). For instance, if we take the LF V=S+I+Y+R, we get \dot V=-\mu_I I-\mu_Y Y \le 0. If \mu_I and \mu_y are strictly positive all equilibria are characterized by (S,0 0,R) and D=1-S-R. So, I don't understand the phrase after (7,8), notice that Y cannot be zero in finite time. For \mu_y=0 then Y* can be nonzero. I guess that closed-form computation of S* and R* is possible as function of the parameters at least in the case v=0. The stability result should be cast in function of the current reproduction number (not explicitated) wrt to S and R. The authors are invited to have a look at

    1.1)Pagliara et al, "Bistability and Resurgent Epidemics in Reinfection Models", IEEE CSLetters, 2018,

    for a theoretical analysis of stability on a similar (just a little bit simpler) model. What said before should be also extended to the stratified model, where a "network" Rt could be defined, see for instance

    1.2)L. Stella et al, "The Role of Asymptomatic Infections in the COVID-19 Epidemic via Complex Networks and Stability Analysis", SIAM J Cont. Opt., 2021, (arxiv.org/pdf/2009.03649.pdf)

    2)It is not clear whether the free contagion parameters of the model have been fitted on real data (identification from infection and reinfection data). Notice that the interplay between vaccination strategies and NPI is important, see e.g. 2.1) Giordano et al, Modeling vaccination rollouts, SARS-CoV-2 variants and the requirement for non-pharmaceutical interventions in Italy", Nature Medicine 2021, where progressive vaccination in reverse age order is considered together with different enforced NPI countermeasures.

    3)In the model the immunity waning is not explicitly considered (flux from R to S or better from a vaccinated compartment to S). It is clear that this complicates the model. Please discuss why the indirect way the waning is considered here is justified.

    3.1)Batistela et al, "SIRSi compartmental model for COVID-19 pandemic with immunity loss", Chaos Soliton and fractals, 2021.

    3.2)McMahon et al, "Reinfection with SARS-CoV-2: Discrete SIR (Susceptible, Infected,Recovered) Modeling Using Empirical Infection Data", JMIR Public health and surveillance, 2020.

    4)Reduction of deaths wrt no vaccination is of course important, but also reduction of stress in hospitals. This is particularly important now with the advent in Europe of the omicron variant. Please discuss on the real message you want to convey to policy makers in the actual scenario of the pandemic.

    Significance

    The final message and some theoretical passages are not completely clear, at least to me. Please discuss on the real message you want to convey to policy makers in the actual scenario of the pandemic.

  4. Version published to 10.1101/2021.11.23.21266766v1 on medRxiv