key: cord-0818462-esdiiihl authors: Koopman, J. S.; Simon, C. P. title: Modeling the dynamics of SARS-CoV-2 immunity waning, antigenic drifting, and population serology patterns date: 2020-09-11 journal: nan DOI: 10.1101/2020.09.10.20192153 sha: 12376b6ec967bab879532f7d7de0636a362df2ff doc_id: 818462 cord_uid: esdiiihl Reinfection with SARS-CoV-2 can result from either waning immunity, a drift in the virus that escapes previously stimulated immunity, or both. The nature of such reinfection risks will affect the choice of control tactics and vaccines. We constructed an SIR transmission model of waning and drifting that can be fitted to cross-neutralization serological data. In this model, waning occurs in individuals who have recovered from previous infections while drifting occurs during transmission to a previously infected individual. Interactions at the population level generate complex dynamics that cause drifting to occur in unanticipated but explainable ways across waning and drifting parameter sets. In particular, raising the fraction of transmissions where drifting occurs slows the rise of drifted strains to high levels and changes the equilibrium distribution of strains from {cup} shaped (extreme strains dominate) to {cap} shaped (central strains dominate). In {cup} shaped parameter regimes, endemic infection levels can rise after many years to above the original epidemic peak. The model simulates results from cross-neutralization assays given sera from previously infected individuals when multiple drifted strains are used in the assays. Fitting the model to such assay data can estimate waning and drifting parameters. Given the parameters, the model predicts infection patterns. We propose a process for using fits of our model to serological and other data called Decision Robustness and Identifiability Analysis (DRIA). This can inform decisions about vaccine options such as whether to prepare for changes in vaccine composition because the virus is changing to escape immunity. Many aspects of SARS-CoV-2 dynamics remain unknown, including the risks of and reasons for reinfection. At least one case of reinfection has apparently been well documented (Kupferschmidt 2020). SARS-CoV-2 reinfections might arise either because host immunity wanes after recovery from infection or because the virus evolves to escape immunity stimulated by prior infections. Reinfection is common and all age groups are repeatedly infected with endemic coronaviruses (Monto, DeJonge et al. 2020 , Nickbakhsh, Ho et al. 2020 . But no studies of any coronaviruses have quantified the roles of immunity waning or antigenic drifting contributing to reinfections. High variation in coronavirus genomes at attachment sites (Andersen, Rambaut et al. 2020) suggest that virus variation might explain some reinfections. When waning of immunity allows for reinfections, those reinfections could generate forces that generate antigenic drift to new virus variants. Such interactions between waning and drifting could also affect the risks of infection after vaccination. To explore such interactions, and to create a basis for extracting information about drifting and waning by fitting models to data, we present a model that captures the separate effects of waning and drifting and their interactions. Our model opens new paths for relating serological data to population patterns of infection and estimating waning and drifting parameters from serological data. The model we examine is simple. Yet it generates complexities we did not anticipate. The understanding we gained in how such complexities emerge will help develop more detailed models with greater capacity to guide both research and control decisions. Accordingly, we lay out a path for building on our model to ensure the validity of scientific or public health decisions. We call this strategy Decision Robustness and Identifiability Analysis (DRIA). Although the model presented here does not include vaccination, adding it is straight forward. It was polio that first led us to develop this model of combined waning and drifting. Besides polio, influenza and pertussis will especially benefit from model elaborations beyond the skeleton we present here. We first describe our model and the behaviors it generates. We then describe how it enables population serological analyses to estimate waning and drifting parameters. Finally, we describe how the DRIA strategy will facilitate decisions about vaccine use. We use a continuous SIR (Susceptible, Infectious, Recovered) compartmental model. The S state represents never-infected susceptibles. The infectious state I is divided into M+1 levels of antigenic drift: I0, I1, …, IM. We write B for the effective contact rate per week between those in S and those in any Ih. An infected individual in state Ih recovers at rate g per week and upon recovery enters recovered state Rh0. The second subscript in Rh0 represents how much the recovered individual's immune system has waned from its maximum effectiveness. As his immune system wanes, the individual newly reinfected with drift level h moves from Rh0 to Rh1 to Rh2 and eventually to RhP (maximal waning) at constant rate w. So, for each drift level h, there are P+1 waning states Rhk, k=0,1,…,P. New infections can occur when an infected in state Ih meets a recovered individual in state Rjk, with the probability of transmission increasing: 1) as the waning level k increases, and 2) with increases in the difference |h-j| between the drift level h of the infected and last former infection level j of the susceptible in Rjk. For example, there is no transmission when an Ih encounters an Rh0 and the highest probability of transmission when an I0 encounters someone in RMP. To quantify this probability of transmission between an Ih and an Rjk, we combine the risk of infection due to waning $ One logic for our choice of this model structure is that drifting of the virus to escape host immunity is a process that takes place during infection and transmission. In an infected individual, diverse viruses are generated. Upon transmission, those viruses that escape a new host's immunity are more likely to cause infection. The model is presented in detail in the supporting material (SM). It makes a number of simplifying assumptions. For example, it assumes that all infections have the same unchanging recovery rate g and the same weekly effective contact rate B (modified, of course, by h,j,k) . It assumes a constant birth and death rate for the whole population, as well as random mixing. It uses a single waning rate w for the P transitions between the waning states. In SM we list other simplifying assumptions that can be realistically relaxed. But we argue that one should not make a model more realistic just to do so because that results in the model becoming less identifiable. Parameter identifiability can be quickly lost even with one or two realistic model elaborations. That is why we emphasize Decision Identifiability which we describe later. We simulated the model delineated in the Supporting Material using the Berkeley Madonna Software (Madonna 2020). For the simulations in this report we set effective weekly contact rate B=1 and weekly rate of recovery g=0.5, so that the underlying basic reproduction number is R0=2. The birth and death rates were set at 1/(75 times 52). All time scales were set to a week. We introduced one infection per 10 million into a continuous population denoted as having 1000 individuals with no immunity or control effects. We worked with 7 drift levels and 7 stages of immunity, so that M=P=6. We varied only the drift rate dr and the waning rate w to understand better the interactions between drifting and waning. Numerical solution of the model used Runge-Kutta 4; the stability of inferences made were evaluated across smaller time steps. Figure 1: Patterns of total infections and infections in the first, third, and fifth drifting states across waning parameters from 0 to 0.1 per week, and drift fraction parameters from 0.01 to 0.5 per week. We have only begun to explore the rich system complexities that emerge from this simple model. We discuss only the interactions between waning of immunity and drifting to escape immunity. These create phenomena that could influence the stability and change dynamics of infectious agents. This simple model needs elaboration to inform theoretical science or public policy decisions. But the dynamics of this simple model are so rich that model elaborations need to proceed with an understanding of what is generating those dynamics. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint The copyright holder for this this version posted September 11, 2020 . . https://doi.org/10.1101 I.2a: A sparse overview across a broad spectrum of waning and drifting To analyze how changes in dr and w affect the dynamics of our system, we ran simulations with seven representative values of each --from no waning and no drifting to high levels of both. We fixed w and let dr vary over these seven values and then fixed dr and let w vary. We present four of these graphs in the SM Figure S1 . Figure 1 gives a sparse overview of these observations. (We will go into more detail below.) In Figure 1 , we vary waning across the columns and drifting across the rows. The dark curve in each subfigure is the graph of the number of infected I(t) from t=0 to t=500 weeks. We also include graphs of I1(t), I3(t) , and I5(t) to understand some of the details of the spread. We present some observations gleaned from or implied by Figure 1 1) Note that the first epidemic is virtually unchanged as w and dr vary in the 12 graphs of Figure 1 , peaking around 153.4 infections at week 32. This invariance is expected since drifting in the model arises from reinfections, which are rare in the first epidemic (nearly all the infected are in I0), and the epidemic ends before waning begins. In fact, up to the 0.1 rate of waning per week, the size of the original epidemic stays the same. 2) Without waning, there is no drifting; drifting can only occur as a result of the immunologically selective forces of reinfection. When w=0 in our model (column 1), epidemics are dispersed in time with virtually no infection between them. They occur when cumulative births generate a large enough (never-infected) susceptible population to raise the effective reproduction number above 1. It takes 75 years for the first epidemic to appear and another 42 years after that for the second epidemic to appear. 3) More waning leads to more infection. In particular, as w rises, so does the timing and peaks of epidemics after the first one. 5) For small values of w, infections occur in discrete waves, with virtually no infection in between. 6) By the time w reaches 0.1, total infection oscillates upwards in time, eventually reaching a positive equilibrium. 7) The size of this equilibrium decreases as the drift fraction dr increases. See supplemental material Figure S1 To provide a feel for the waning parameter values in Figure 2 , we present in row 2 of Table 1 the waning levels in terms of the time it takes for half of recently infected individuals to lose all of their immunity. This is the time when half the previously infected population is at waning level 6 (the sum of Rj6 across j). In Row 3, we present the waning levels in terms of the time for the sum of all susceptibility across the whole population to reach a level that has half the immunity it had right after infection. Figure 2 presents the prevalence of infections at different drifting levels. As in Figure 1 , the first epidemic is unchanged by the waning or drifting parameter values. However, at each positive waning level, the timing, size, and drifting level composition of the second and third epidemics changes as dr increases from 0.01 to 0.5. At waning level 0.01, the period between these first two epidemics shortens gradually as the drifting parameter increases to dr=0.03, after which it lengthens again. The second and third epidemics are increasingly populated by more drifted viruses as dr is increased up to a value of 0.1. But by drifting fraction 0.5, there is a marked change as the second and third epidemic infections are mainly at drift levels zero and one. At waning level 0.05, the second epidemic appears quickly and subsequently there are epidemic waves rather than distinct epidemics. This is even more the case for waning level 0.1. For example, consider the case w=0.05, dr=0.01 in Row 1, Column 2. We blow up this subfigure in section 2.1 (Page4) of the SM Figure S1 . As seen in this supplementary figure, after the initial . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint The copyright holder for this this version posted September 11, 2020 . . https://doi.org/10.1101 epidemic, the total infection fluctuates upwards through a series of six waves. As usual, the I0s dominate the initial wave. But then, as seen in Figure 2 , I1 dominates the next wave, I2 the next, I3 the next, ..., I6 the last, after which total infection eventually equilibrates at ≈ 56.5/1000. Throughout this process, because of the small drift fraction 0.01, I0 never goes away. In fact, it is the second most active drift level in each wave after the first. Its encounters with neverinfected susceptibles further sustains the numbers of these persistent I0s. The resulting R0ks have the highest probability of reinfection when they encounter each new Ij because 0 is the furthest index from j. Soon after the I6s arise, the I0s and I6s become the dominant strains. All this required a low drifting fraction. As the drift fraction dr increases, the role of the I0s decreases. When dr =0.1, each Ih still plays a major role in wave h, but each becomes more persistent over time. When the drifting fraction reaches 0.5, the impact of the early drift levels I0 and I1 diminishes and the intermediate levels, especially I3 eventually take over. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint The copyright holder for this this version posted September 11, 2020 . . https://doi.org/10.1101 There is less drifting to the extremes as the drifting fraction parameter dr is increased because: 1) There is less amplification of the extreme drifting levels since the fraction with potential to stay at the same level is 1 − , which decreases as dr increases; 2) There is more drifting away from the high and low extreme levels as the dr increases, 3) As the infection level goes down for the first two reasons, there are more uninfected individuals that have an absolutely flat susceptibility to infections at all drift levels. That further reduces the forces driving drifting to the highest and lowest drifting levels. By the end of week 500, drifting levels are converging to equilibrium levels in Figure 2 . Figure 3 presents the numbers of infective of each drift level when the system finally equilibrates. Extreme levels I0 and I6 dominate for small dr, intermediate levels I2, I3, and I4 dominate for dr=0.5. Our model assumption that transmission probabilities are highest when the strains are furthest apart leads to higher infection levels for smaller dr's. Waning level patterns are a determinant of reinfection potential. Figure S2 in SM tracks the waning levels for the parameters in Figure 2 . Waning cascades from level 0 to 6 where it accumulates. The population level force of infection is a major determinant of what waning levels dominate across time. At the very low waning rate of 0.01 per week, waning cascades slowly and never accumulates much in level 6 except at the very high drifting fraction. The dual role of the drifting fraction dr in initiating waning more quickly but causing less growth after initiation explains this pattern. At the higher waning rates there is more accumulation in w6 (Max Wane) during the first four years. But then as the drifting levels of viruses drift apart and cause an increasing force of infection, w6 is drained by new infections and these cause increasing levels of w0 that cascade down. Equilibrium levels of w6 decrease thus decrease more than lower waning levels. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint The copyright holder for this this version posted September 11, 2020 . . https://doi.org/10.1101 We explored different model formulations where drift levels went in circles rather than ending at the extremes. These formulations produced similar results. See SM 2.3. We have demonstrated interactions between waning and drifting that affect the population patterns of infection. The patterns produced by these interactions provide a way to estimate waning and drifting parameters by fitting the model to observed data. But we don't want to wait many years for patterns to emerge like those seen in figures 1 & 2 to estimate those parameters. We want a methodology that can use the serosurveys currently being conducted to help make decisions quickly regarding how the challenges of waning and drifting should be addressed through vaccination programs. Those serosurveys include 1) prospective follow up of infected individuals to estimate waning as a function of time since infection, and 2) serial cross sectional samples of sera to characterize population patterns of infection. The methods we present in this section will efficiently use the second sets of data to describe the combined effects of waning and drifting. Then using the DRIA approach we will present in the next section, vaccine decisions can be informed in a more timely and valid manner by using both types of data. How model generated data is turned into cross-neutralization analysis data Our model generates tables of cross-neutralization data. The real world, through the work of adept serologists, also generates tables of cross-neutralization data -with the same information. That means that we can adjust model parameters or otherwise modify our model until there is correspondence between our model and the real world. To do that well, we first need to understand how our model generates the data and how changes in model parameters alter the data. Providing the needed understanding is the goal of this section. In the SM we provide a tutorial on interpreting population cross-neutralization data (SM3.1 page 9). We give a brief overview here. Suppose there are V virus types under consideration; they may differ, for example, by strain, year, or location. Choose a sample of uninfected individuals and take a blood sample of each to measure neutralizing antibody in the serum. Dilute each serum by a fraction T times for each individual in the sample. Assay each dilution to see if the serum neutralizes the growth of each of the V virus types. Record for each virus type the most diluted serum that neutralized it. This measure of immune response to each of the V virus types yields an entry in a × ⋯ × (V times) table. Then count the number of individuals whose entry fell into that slot to generate a cross-neutralization table. We use neutralization assays as the serological method only because neutralization is widely thought of as being the most direct assessment of immunity. But more informative assays can be developed and related to immunity using the DRIA process to be discussed later. In the case of V=2 virus types, if all the entries are on the diagonal, then the viruses are equivalent with regard to drifting status. On the other hand, if there has been drifting so that . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint The copyright holder for this this version posted September 11, 2020. . https://doi.org/10.1101/2020.09.10.20192153 doi: medRxiv preprint each viruse can escape some of the immunity stimulated by the other, table entries will be on the sub-or super-diagonals. Our model has 7 virus drift levels Ih and 49 categories Rjk of previously infected individuals (7 drifting levels j by 7 waning levels k). To measure from our model output the neutralization level of virus type Ih by the immune system of an individual in an Rjk, we use the transmissibility measure Z(h,j,k), as formulated by function (1.1) in our model (without the B factor). In effect, the Z function corresponds to the inverse of the neutralizing level. To make that number correspond to a titer, we divide the interval [0,1] into 10 equal subintervals that could correspond to 10 sequential dilutions for the neutralization assay. Both waning and drifting determine the neutralizing antibody levels according to equation (1.1). In this exposition, we work with two or three virus types at a time. The cross-neutralization table for 2 viruses is a 10 by 10 matrix. To get a particular value for each of the 100 entries in this cross-neutralization matrix, we first determine a susceptibility level for each of the 49 Rjk subgroups. Then we sum up the total number of individuals in the population that have horizontal axis level of susceptibility in the table corresponding to h1 and the vertical axis level of susceptibility corresponding to h2. Each entry thus has identical people with regard to neutralizing antibodies but it can be composed of individuals with different waning and drifting levels. To describe this process more analytically, let Π(z) denote the subinterval to which z in [0,1] belongs; analytically. Π(z) is 1 + the integer part of the decimal expansion of 10·z. Then, the (m,n)th entry of the cross-reaction matrix is the total number of individuals in all the compartments Rjk for which Π(Z(h1,j,k)) =m and Π(Z(h2,j,k)) =n. We numerically solved the model after introduction of 1 case per 10 million population and took cross sections of the uninfected population at 60, 110, and 162 weeks after that introduction. In the SM section 3.2 we examine the cross-neutralization patterns produced by four drifting fraction parameters, two waning rate parameters, and cross-neutralization using viruses at drifting levels 0 and 1 for all three time points and levels 0 and 2 for times 110 and 162. During the first three years, Figures 2-3 and S2 show small or no differences between patterns with different drifting fraction parameters but the same waning level. The most valuable information is not captured in those figures because they do not break down the previously infected population by both drifting level and waning level. That breakdown is where the information on drifting lies. Cross-neutralization titers can perceive that breakdown, as can be appreciated across the range of parameters examined in Figure S3 . For illustration purposes we present 60, 110, and 162 week cross-neutralization patterns for drift levels 0 and 1 at 60 and 110 weeks and 0 and 2 at 162 weeks given a waning rate of w=0.1 and a drifting fraction of dr=0.1. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint The copyright holder for this this version posted September 11, 2020. . https://doi.org/10.1101/2020.09.10.20192153 doi: medRxiv preprint Figure 4 : Expected cross-neutralization assay results given model parameter values for waning rate = 0.1 and drifting fraction = 0.1 with a population cross sections taken at 60, 110, and 162 weeks and viruses at drift levels 0 and 1 used in the earlier assays and levels 0 and 2 are used at 162 weeks. Note that blank cells had no population in them. Those with zeros had less than 0.05 population. All of those with the lowest titers on both were never infected individuals highlighted in yellow. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint The copyright holder for this this version posted September 11, 2020 . . https://doi.org/10.1101 At most parameter settings there was little evidence of drifting at 60 weeks. At 60 weeks 25 out of 628 individuals with a titer greater than level 5 had a higher titer to the strain at drifting level 1 than to the strain with a drifting level of 0. But by week 110, 351 out of 667 did. In the third panel of Figure 4 we see how at 162 weeks, drifting to level 2 and very slightly beyone is detected. It also shows how intermediate levels of drifting might appear when there has been further drifting from an initially drifted virus. Note that this third panel uses drifting levels 0 and 2. If the world corresponded to this oversimplified model, if the world was deterministic like our model rather than stochastic, and if we could have measured the world perfectly just like we measured our model output, a decision at 60 weeks to adjust the vaccine might have been made. Using the DRIA approach to be discussed in the next section, we don't need our model to perfectly correspond to the real world or our measurements to be perfect in order to make a good policy decision. But we do need real world data and we almost certainly would need to realistically relax many of the simplifying assumptions in our model. Additionally, we might need to insure the identifiability of our decision with additional data like that discussed in SM section 4.4. But serology data should be a great asset to making a valid decision because our model can generate it and thus the parameters of the model can be estimated by fitting the model to the data. Note that by making inferences about what fraction of the population has been infected by drifted or undrifted strains, cross-neutralization assays not only provide information about drifting and waning parameter values. They also make inferences about what fraction of the population has been infected with differently drifted strains. This information, together with waning and drifting parameter values, and a DRIA process that ensures robustness and identifiability is what enables our model to predict the future. The SM in section 3.3 considers additional things that can be gleaned from the crossneutralization data and explores issues of why the data turns out as they do. The biggest determinant of the shape of the cross-neutralization data is the past history of infection. Higher waning leads to more reinfection, which leads to more drifting, which in turn leads to more reinfection. There can be no perfect separation the waning influences from the drifting influences. But drifting effects show up in the off diagonal cells and are dependent upon having viruses in the assay where genetic changes have immunological effects. One can improve their feel for model behavior by seeking to understand why the shape of the cross-neutralization patterns changes as parameter values change in terms of the past history of the population. To get the full information out of population level cross-neutralization assays, one must fit a model to actual cross-neutralization data from appropriately designed sero-surveys. One gets that information out by treating the model as a partially observed Markov process to fit the model to data (Funk and King 2019). . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint The copyright holder for this this version posted September 11, 2020. . https://doi.org/10.1101/2020.09.10.20192153 doi: medRxiv preprint The model we have presented is a first step in a process needed to make valid scientific or policy decisions. The DRIA process we propose, as shown in the figure below has a number of distinct steps that will make subsequent investigations more productive. The enumerated steps are as follows: 1) formulate the decision to be made in terms acceptable to policy makers, 2) construct a simple model like the one in this paper, 3) fit the model to data, 4) assess whether a decision is identifiable given the data used. This requires two tasks to be completed. First, decision boundaries must be mapped out in parameter space. Second, the model must be fit to data and the parameter space that is consistent with the data must be mapped out. This key step is addressed by using one of the modern approaches to fitting the model to data (Funk and King 2019). A decision is identifiable when the parameter space consistent with the data falls entirely into one decision parameter space or another. 5) If the decision is identifiable, then one proceeds with inference robustness loops that put the decision on firmer grounds because it is less likely that unrealistic model assumptions could be determining the decision. 6) If the decision is not identifiable one proceeds to a decision identifiability loop and seeks more informative data or model changes which better use of available data. These make the decision more identifiable. These steps are taken in an iterative process involving two loops as shown in Figure 6 above. Those loops address the two major sources of errors when using models to make decisions about complex dynamic systems: 1) Some aspect of model structure that does not correspond to reality leads to a wrong decision. 2) The fit of the model to data which leads to a decision is not the only fit possible and other fits could lead to a different decision. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint The copyright holder for this this version posted September 11, 2020. . https://doi.org/10.1101/2020.09.10.20192153 doi: medRxiv preprint Decision robustness loops can add model complexity until identifiability is lost. Then even if new data is found, more inference robustness loops can put one back into identifiability loops. If one is pursuing a decision about a scientific theory formulation, decision robustness and identifiability loops could go on infinitely. Policy decisions, in contrast, must be made before things get out of control. Therefore, they require judgements that could be improved by collaborations between scientists and responsible administrators. A decision about scientific theory that DRIA should address is whether immunity driven drifting in pandemic SARS-CoV-2 strains are sufficient for it to sustain transmission through reinfections. This scientific decision should help later focus on public health control decisions related to drifting. For example, a decision is needed as to whether to invest the many billions of dollars in vaccine development that can handle antigenic drifting as influenza vaccines do. IV. Discussion of how this model can improve pandemic coronavirus control This model opens three new ways to improve pandemic control. 1) It provides a framework to think about the causal systems where waning and drifting contribute to repeat infections. 2) It provides a path to use serological data to inform transmission system analyses more fully. 3) It provides a way of observing whether the way we think about interactions between waning and drifting is occurring in the real world by fitting the model to serological data. In our analysis of our simple model, we observed concerning levels of drifting after one year only at high levels of both drifting fraction and waning rate parameters. At lower parameter values we still observed notable drifting over longer intervals. Our analysis, however, requires fitting the model to data and following out DRIA procedures before any action inferences are justified other than gathering the data and carrying out the analysis. Hopefully we will have a vaccine soon and our major concern by the time significant drifting has occurred will be assessing whether the virus can escape not only the immunity it stimulated, but also the immunity provided by vaccines. The immediate use of this model should be to guide the collection and analysis of serological and epidemiological data that, together with further model elaborations, should inform decisions about future waning and drifting risks and their effects on vaccine choices and vaccine administration strategies. Andersen, K. G., A. Rambaut, W. I. Lipkin, E. C. Holmes and R. F. Garry (2020). "The proximal origin of SARS-CoV-2." Nat Med 26(4): 450-452. Funk, S. and A. A. King (2019). "Choices and trade-offs in inference with infectious disease models." Epidemics 30: 100383. Kupferschmidt, K. (2020). "Some people can get the pandemic virus twice, a study suggests. That is no reason to Panic." Science 369(6507). . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint The copyright holder for this this version posted September 11, 2020. An infected individual in state Ih recovers at rate g per week and upon recovery enters recovered state Rh0. The second subscript in Rh0 represents how much the recovered individual's immune system has waned from its maximum effectiveness. For each drift level h, there are P+1 waning states Rh0, Rh1,…,RhP. At each time step a fraction w of those in any Rhk (kh and h' = h-1 if jJ-1. In the third line, new IJs arise through drifting when IJ+1s transmit to Rhks with h