Epidemiology: Gray Immunity Model Gives Qualitatively Different Predictions

Epidemiology: Gray Immunity Model Gives Qualitatively Different Predictions Milind Watve*, Himanshu Bhisikar, Rohoni Kharate 1 Independent researcher, E-1-8, Girija Shankar Vihar, Karve Nagar Pune 411052, India. milind.watve@gmail.com 2 Indian Institute of Science Education and Research, Homi Bhabha Road,Pune 411008, India. himanshu.bhisikar@students.iiserpune.ac.in 3 Independent researcher, 4, Shivratri Soc., Model colony, Shivaji nagar, Jail road, Nasik, 422101. rohinikharate1712@gmail.com


Introduction:
Mainstream epidemiological models use a compartmental approach in which the population is dynamically distributed into compartments, typically three compartments namely susceptible, infected and removed (Tang et  The compartmental models consider immunity as a binary variable. Any individual or a fraction of the population, when not suffering from active infection is either susceptible or immune in the model. We show here that considering immunity as a continuous variable rather than as a binary variable and incorporating subtle known factors affecting immunity explains many of the previously unexplained features of the epidemic. Allowing gray shades of immunity one can get repeated waves without involving new variants, increased rates of transmission without increased infectivity of the virus, a long term endemic steady state, a peak and decline much before the expected herd immunity threshold, breakthrough infections after vaccination and new surges after vaccinating majority of the population. The model that can explain these phenomena also gives certain non-conventional predictions that can be potentially important for designing control measures for future epidemics.
Incorporation of shades of immunity and dynamic population distribution of it necessitates an individual based model where the immunity level of each individual is affected by a number of factors. We only incorporate factors for which there is empirical evidence, at least qualitatively.
Infection of an individual is a probabilistic phenomenon. In our approach immunity is measured in terms of the dose of infectious agent required to infect an individual with a probability of 0.5. Higher the level of immunity, higher is the ID50, i.e. higher the dose of pathogen required to cause infection. An individual's ID50 is a dynamic variable of the model that can vary between zero to infinity and that is under continued flux being affected by a natural infection or vaccination, which leads to a large quantum jump in ID50. This contrasts with many small immunity modulators. Some immunity is contributed as cross immunity from infection by other After recovering from a respiratory infection, for example, much of the mucous membrane is composed of young cells that have replaced the infected cells. Such a young cell lining may be more resistant to a new virus, but this effect is expected to be short lived and wane fast as some older and effete cells accumulate. It is also possible that exposure to a pathogen at sub-infectious level contributes to some immunity (Gold et al 2021, Santos Rocha 2018, not comparable in magnitude to that achieved by active infection or by vaccines (Krammer 2021). Since immune response is costly, the body has evolved not to give a full strength immune response to every attack. Immune response is proportional to the intensity of invasion and virulence of the pathogen (Read 1994, Long et al 2020, Shinde et al 2021. Therefore the response to a mild exposure without detectable infection is likely to be proportionately small, but repeated exposures may raise the immunity level sufficient to increase the probability of escaping a given attack. In the absence of repeated exposure, immunity is known to decline gradually (Leino 2000, Sanderson 2021). This phenomenon was well recognized by previous models, but they still treated it with the binary immunity state assumption (Leung et al 2018). Different components of immunity are known to decline at different rates. For example, memory cells have a long life (Turner et al 2021) whereas antibody titres decline relatively fast. Some of the other non-specific small contributors mentioned above may also decline rapidly. At present there is little empirical work to parameterize these subtle factors. Therefore we do not incorporate the differential decline, but assume a small constant decline in ID50 with time in the absence of repeated exposure. The results of accommodating the small immunity effects (SIEs) are qualitatively different from the binary immunity models.
Owing to lack of empirical data on all the immunity related parameters, it is not possible to make a quantitatively predictive model for the Covid-19 pandemic at this stage. But our objective is to demonstrate that the SIEs can substantially affect the shape of an epidemic curve and therefore they demand more empirical inputs as well as a different class of futuristic models that will have a greater predictive value.

The Model:
We conceive an individual based model in which every i th individual in a population of N has a dynamic immunity level ID50 (i,t) which can change with time. There is a background population distribution of immunity levels before the epidemic begins which is assumed to be normally distributed initially with a mean ID50 (0) and s.d.ID50 (0). The nature of the distribution is allowed to change in time as individuals change their immunity levels. The probability of infection at a given level of exposure to the pathogen is assumed to be a sigmoid function by the classical principle of ID50 such that Where ( ) is the exposure bias (in arbitrary units) of i th individual to the virus. Individuals differ in their exposure to a pathogen by their profession, locality and behaviour and accordingly E(i) also has a population distribution which is assumed to be constant in time. The parameter 'a' is the power that decides the sharpness of the sigmoid curve. We assume a small chance Pcross with which an individual is infected by other viruses contributing cross-immunity. The level of cross immunity offered by such an infection is Icross assumed to be one to two orders of magnitude smaller than Iinf. Similarly we assume that an exposure to the infectious agent without causing infection also confers a small immunity increment Isce and this also works in a similar sigmoid function with Sce50(i) corresponding to probability of 0.5. Sce50(i) also has a population distribution which has a constant mean and s.d.
Whenever an individual's status is 0 and remains so, with neither Icross nor Isce increments, the immunity is assumed to decline by a small decrement Id which is much smaller than Isce and Icross.
The exposure of individuals to the infectious agent also has multiple components and we include three important parameters representing them. One is the exposure bias of an individual. Some individuals by their occupation, type of housing, mobility and behaviour are more exposed and others relatively isolated. Therefore E(i) is assumed to be the distribution of the exposure bias Similarly the probability that an individual gets a subclinical exposure that gives a small immunity increment Isce at time t+1 is We used stochastic simulations using these probabilities. The range of parameters used for these simulations is tabulated in Since empirical estimates for many of the parameters are not available, we restrict ourselves to drawing qualitative conclusions that show alternative possible outcomes of any policy or measures implemented and do not claim any quantitative predictions. We expect NPIs to affect mean E(i), Pcross, Kcmax and K to be affected by NPIs but in a differential way. Personal protection measures such as masks, PPEs, hand washing, surface sanitization and social distancing is expected to reduce meanE(i) and Pcross. Travel restrictions and work from home policy are expected to decrease Kcmax and/or increase K. We also consider a general lockdown parameter L that goes from 0 to 1 and which multiplicatively alters E(i), Pcross and Kcmax simultaneously and proportionately. These changes were applied either from the beginning of the simulations or at different t representing imposition or relaxation of NPIs at different stages of the epidemic.

Results:
Since the model involves a large number of parameters we started with an exploratory approach to randomize all parameters within the given range (table 1) and record the qualitative outcome.
Over a thousand simulations gave only four qualitative types of outcomes (figure 2),(i) A single peak followed by extinction or near extinction of the pathogen, similar to a typical compartment model (ii) a peak followed by a low level stable or mildly fluctuating endemic like state (ii) a pattern of multiple surges or waves, the surges often being separated by apparently stable or fluctuating incidence for variable time duration (also see figures 7, 9 and 10) for variable spacing between waves). Since the number of parameters is large and we have no empirical estimates for many of them, we do not systematically explore the entire parameter space. Instead we focus here on demonstrating the complex interplay of parameters, the overall complexity of the outcomes and the inherent unpredictability of the system although certain qualitative pattern predictions can be made. This contrasts the classical deterministic predictive models and the change is brought about by making just one binary variable continuous.  When the small immunity effects are zero or close to zero, the outcome is similar to classical compartment models implying that the apparently stable endemic or multiple well spaced wave patterns are a result of the SIE. It is notable that in this model when SIE parameters > 0, the height of a peak and subsequent decline begins when only a small fraction of the population is infected, in contrast to the herd immunity threshold of classical SIR model. An important outcome of the model is that a wave pattern with variable spacing, slopes and heights of the waves is possible even without the need for novel variants defying immunity.
Although the system assumes a state of complexity with limited quantitative predictability, the processes leading to the four possible outcomes can be explained in principle. Owing to the sigmoid relationship of infection probability with the dynamic individual ID50 values, at times a small increase in immunity can be sufficient to evade infection with a high probability. The escape is likely to be accompanied by further small increase in ID50 reducing the probability of infection further. By this mechanism, many individuals can escape infection without being "fully" immune. This results into an arrested peak followed by decline in incidence much before the classical herd immunity threshold could be reached. One of the patterns predominantly observed during the first phase of the pandemic was that most members of the family of infected individuals appear to have escaped infection in spite of exposure (Shah et al 2020). This can also be explained by the SIE effect. However, as the incidence declines, the immunity levels achieved by SIEs also start declining. The rate of decline is enhanced by NPIs. This makes some of the individuals susceptible again. This leads to a complex dynamics resulting into a fluctuating incidence that may remain apparently stable for some time or give rise to another surge when the immunity levels of a substantial part of the population decline below a threshold.
The immunity decline is more relevant to individuals with smaller E(i) as they remain protected owing to lower exposure. However, as they remain protected, their immunity also declines slowly. With the decline, they become increasingly susceptible at a lower exposure. This is unlikely to happen to individuals who achieve immunity after an active infection or by vaccination. This is because the rise in immunity is of a much greater magnitude than the SIEs.
Although the infection or vaccination acquired immunity is also subject to waning, during the long time required for waning they have a greater chance of repeat subclinical exposures boosting the immune levels again. Therefore the second wave is mostly due to decline in the SIE rather than decline in infection or vaccination driven immunity. This is also evident in the gap between two surges which can be often much smaller than Iinf/Id.
When the rate of a process is determined by multiple factors, only some of them are rate limiting at a given set of conditions. Therefore only intervention in these factors can result into effective   We see in the Covid-19 data across different populations that most of the peaks have been much smaller than the herd immunity thresholds that were predicted. This is compatible with our model. However there are two possible alternative explanations for having dwarf peaks. One is that the non-pharmaceutical interventions (NPIs) or preventive restrictions (PRs) effectively restricted the transmission and arrested the peaks. The other is that most individuals could escape infection owing to the SIE effects restricting the peak height. It is possible to make differential predictions from the two alternative hypotheses for the small peaks. If the preventive restrictions arrest the infection and turn the curve downwards by effectively making R < 1 in the classical model, the slope of the downward curve is expected to be independent of the upward curve (figure 5a). On the other hand, in the classical model, as well as in our model when the curve naturally starts a decline owing to altered level of population immunity, there remains an element of symmetry in the shape of the peak. If the upward slope is steeper, the downward slope is also proportionately steeper. This is because a rapid rise in incidence also causes a rapid rise in immunity by the classical model as well as by SIE effects and a greater level of population immunity causes a steeper decline. This leads to a good correlation between the upward and downward slopes as revealed by simulations ( figure 5b and c). Applying PRs at some point in a rising wave, and assuming that the PRs are effective, a decline before the herd immunity levels can also be obtained. But in this case the slope of the decline is driven by the intensity of the PRs and therefore may not be related to the upward slope before the imposition of PR. High population density and other pro-transmission factors that are responsible for a steep upward slope, make a steep downward slope more difficult. This is most likely to deviate further from the symmetry of the wave. Therefore using a correlation between the upward and downward  The starting ID50 distribution in the population is assumed to be normal in the SIE model. However, the shape of the distribution will change as the epidemic progresses. While infected and recovered individuals will have experienced a quantum jump in their ID50(i,t) the protected part of the population may have lost some, particularly if the personal protection measures are strictly followed. Further as a wave recedes, exposure becomes rarer and the immunity is gradually lost for a larger sector of the population. The distribution of immunity in the population at this stage is highly dispersed and often bimodal ( Figure 6). Although the individuals having acquired immunity after an active infection will experience little proportionate loss and will be least susceptible to reinfection for a longer time, the once having escaped because of SIE may have reduced immunity as compared to their baseline particularly if personal protective measures have been strictly followed. The resulting bimodal or over dispersed distribution of immunity has important consequences for the shape of the incidence curve. The pathogen may persist in the population without becoming extinct owing to newly created susceptibles that can get infected at very low exposures. If the immunity of a substantial sector of population drops below a threshold, a new wave may begin. Therefore depending upon the parameters and the standing context, the epidemic can take wave forms without the need for new variants or behavioural change in the population. The possibility of variably spaced repeated waves is a unique outcome of the SIE model not seen with the classical compartment models without involving novel variants or recruitment of new born susceptibles.

The immunity levels decline starting with the protected sector of the population (low E(i)) and
when near the lower end, some of them get infected even at very low exposures. These individuals are thrown to the right extreme because of infection induced immunity, but simultaneously more individuals lose immunity to take their place. This immunity dynamics appears to be mainly responsible for the stable or oscillating incidence.
Nevertheless the model is not incompatible for new variants which might escape the specific immunity to earlier variant or may be mutants with higher infectivity. It is possible that both SIE and new variant contribute to repeated waves. In the context of repeated waves, another interesting result of the SIE effect is that even in the absence of a new and more infectious variant, the second wave can be steeper and taller than the first one ( figure 7). This happens when initial population immunity, mean ID50(i,0) is substantially greater than mean E(i), both the distributions have sufficiently large standard deviations and personal protection or general lockdowns are sufficiently effective to allow loss of immunity in a sector of the population.
Under such conditions the first wave is mainly limited by pre-existing cross-immunity of the population. However, due to personal protection measures exposure to other viruses giving cross-immunity is also reduced for the majority of the population. As a result the distribution of immunity becomes bimodal and for substantial population the immunity declines below a threshold to trigger a new wave. Since now the background immunity for a sector of the population is lower than the first wave, the second wave rises more sharply and achieves a greater height. Such a pattern is seen in many countries during the current pandemic. This creates an alternative possible explanation to the more devastating second wave faced by countries such as India. The first wave in the highly population dense India was said to be surprisingly small with low mortality which could be ascribed to a background immunity level contributed by other    This contrasting pattern is testable in real life. Although precise quantification of individual exposure bias may be difficult, it can be predicted that if SIEs cause a first dwarf peak followed by a second wave, individuals whose occupation exposes them more to infected individuals, who live in population dense locality and/or do not follow appropriate behaviour will be infected disproportionately more in the first wave whereas in the subsequent waves individuals from more isolated, remote and low density areas, with safer occupations and following Covid appropriate behaviours will show disproportionately increased relative incidence.
The bimodality of immunity in the SIE model also suggests a possible cause of the observed incidence surge in many countries after nearly half or more of the population getting vaccinated.

Although the mean immunity of the vaccinated sector goes up (dotted green line) pushing the average up (red line), the unvaccinated ones continue to lose immunity slowly (dotted violet line) and after a threshold loss a new surge may begin.
The model finds its most important use in predicting the qualitatively different possible outcomes of PRs. The generalized outcome is that the effects of PRs can be non-monotonic and depend substantially on the context. What can be good in the short run may turn counterproductive in the long run ( figure 10a). Also the stringency of PRs may not be linearly or even monotonically related to the total incidence (figure 10b) as observed commonly in simulations over a large parameter space.  Since we have little information about the SIE related parameters, it is difficult to predict whether PRs will be beneficial or hazardous in different populations at different times.

Discussion:
An epidemic disease is a complex system which is difficult to predict quantitatively as witnessed during the ongoing Covid-19 pandemic. The predictions of the current mainstream modelling based on compartmental models largely failed in making qualitative as well as quantitative predictions. Not only the number of cases or number of deaths could not be predicted, but that there would be repeated waves, the causes, the timing and the severity of waves also could not be predicted at a level significantly above common sense. This is likely to be an effect of oversimplifying a complex system. Modelling is best done with an incremental approach. When a simple model fails to meet its objectives, it is necessary to incorporate crucial elements of the complexity into the model. What we show here is that treating immunity as a binary variable Of particular interest is that the model gives a possible alternative causal hypothesis to the dwarf peaks and repeated waves which are currently believed to be caused by differential implementation of PRs and by new variants respectively. The three factors are not mutually exclusive and could be acting together or in complex interactions. In the light of the gray immunity model it is possible to take a perspective of an analytical as well as accommodative causal analysis. There are two possible reasons for the dwarf peaks. The currently believed one is that the PRs restricted the peaks and the alternative explanation is that most individuals escape infection owing to small immunity increments. We have already stated the differential predictions of the two. According to the PR hypothesis, there should be no correlation between the upward slope and downward slope of the peaks which is expected to be quite strong by the SIE hypothesis. In reality most peaks are highly symmetrical as shown by the highly significant correlation between upward and downward slopes. In addition, the analysis of change in slope of the incidence curve following imposition or relaxation of PRs has already shown that the effects In case the new variant is only partially susceptible to prior acquired immunity, the binary immunity model is inappropriate to accommodate this possibility and our model needs to be brought in. The moment the continuous immunity model is invoked, it brings in its own intrinsic pattern of repeated waves, which should form the null model against which the new variant hypothesis should be tested. Since new variants keep on arising and their relative frequencies may drift or get selected by any mechanism of competition between viruses, the association of a variant with a new wave could only reflect a coincident hitch hiking on a wave. This null hypothesis can be rejected if a consistent correlation is shown between the increasing frequency of a variant and increasing R of the incidence curve across different countries or populations.
Awaiting such a critical testing of the new variant hypothesis, at present both the alternative (but not mutually exclusive) hypotheses need to be kept open. It is also possible that a new variant gets selected by the altered immunity landscape of the host population. Since the immune response of a host is proportional to the intensity of invasion, it is likely that more invasive variants get selected when the host immunity is low (Shinde et al 2021). Therefore SIE is also likely to be causal to selection of more infectious variants. A critical question to ask is whether appearance of new variants is mutation limited or selection limited. It is possible to address this question with retrospective data. If it is mutation limited, we should see most new variants appearing near the peak of waves when viral populations are at their maxima. If selection limited, extended NPIs will be associated with more infective variants. Thus NPI, selection on new variants and small immunity effects are likely to be intertwined threads whose effects are difficult to segregate from each other.
Apart from the academic implications of our model in rethinking of the modelling approach, there are direct public health implications as well. The possible trade-off between short term and long term effects of PRs and the non-monotonic outcomes caution against blanket recommendation of lockdowns as well as personal protective measures over a long term. The measures that have a large social and economic cost should not be recommended since their outcomes are context dependent and at times turn counterproductive. Where and when it will turn counterproductive cannot be predicted very well at present. Therefore umbrella recommendations of such measures should not be done. In fact, this was the stand taken in a WHO report published a few months before the beginning of the pandemic (WHO 2019). But the cautionary note against implementation of the socially and economically costly NPIs appears to have been forgotten under the panic response to the pandemic. After vaccination, repeated exposure is most likely to boost and maintain long term immunity (Leino 2000). Therefore according to our model, other personal protection measures after vaccination should be contraindicated. At least substantial rethinking is required about public health policies to control infectious disease epidemics. One major hurdle in this has been lack of empirical studies on the small immunity effects, which is a hen and egg problem. Since the importance of SIEs is not appreciated, there is little motivation for empirical studies and since there is no data, modelling involving them fails to progress beyond a limit. At present we don't even have tools to monitor the small immunity changes at a population level. Antibody titres do not reflect all mechanisms of immunity. Population screening tools for other subtle mechanisms have not been developed.
Therefore they are not incorporated in the current epidemiological thinking. We have tried to break this vicious cycle by indicating that at least theoretically the SIEs can alter the course of an epidemic substantially. The change is fundamental because what is currently assumed to limit the incidence may possibly be increasing it in the long run, as suggested by the model. At the minimum, the message of the gray immunity modelling exercise is that the factors assumed to be small and unimportant need more attention since they can potentially change public health policies fundamentally.