Corona pandemic: future considerations from the perspective of epidemiological modelling

In the simplest case, so-called SIR models are used in the mathematical modeling of epidemics. The three capital letters describe the states of the individuals or population relevant to the infection process: S means being susceptible (susceptible) to the infection, I means being infected and infectious, and R stands for resistance (immunity). As elsewhere, SIR models are predominantly used at the Robert Koch Institute (an der Heiden 2020; Maier 2022). Without further modifications, SIR models are used for modeling infectious diseases, which leave lifelong immunity after recovery, as is the case with many "childhood diseases". But this does not seem to apply to an infection with SARS-CoV-2. There is evidence that both infection with the coronavirus and vaccination against the coronavirus produce only temporary immunity (Baraniuk 2021). In addition, the virus can circumvent an existing immunity with new virus variants (so-called escape variants), against which cross-immunity from previous infections is not available or insufficient (Starr 2021; Weisblum 2020). This article neither describes the incidence of the disease in society nor the burden on the healthcare system. Instead, modeling will be used to examine how various factors could influence the transmission of the infection. In this case, infections must not be equated with diseases that are influenced by many other factors (age, previous illnesses, etc.); infections that occur after previous vaccinations or after previous infections are also associated with a different probability of illness, which is very likely lower. The article presented here focuses on the parameters relevant to infection epidemiology, in particular concerning prevalence and immunity.

The special feature: loss of immunity

>> If corona infections generate only temporary immunity, this implies a continuous loss of immunity in the population. People who have already recovered from an infection with the virus can become susceptible again and subsequently contract the coronavirus again (or more often) (Pulliam
2022). This is described by so-called SIRS models, which take into account the possible “relapse” from the immune to the susceptible state (R -> S). If you also take into account that infected people do not immediately become infectious, but first go through a latent period, you get a SEIRS model (E stands for “exposed” in international usage and refers to the latent period between the infection and the infectious period). .
The model predictions of a SEIRS model compared to those of a SIR model are shown schematically in Fig. 1. The key difference is that a SEIRS model leads to the endemic state quite quickly, whereas a SIR model typically only produces a single epidemic and can only regain a foothold in the population once many of the immune ones have died and passed through newborn susceptibles were replaced. The most important difference between the two models is how and how quickly susceptibles arise in the population: in the SIR model they only arise through the birth of children; in the SEIRS model they also arise through loss of immunity.
If immunity after infection is only temporary, there is inevitably a risk of an endemic condition, which can also cause recurring epidemics. This risk increases when there is a seasonal variation in transmission rates, such as in: B. has been described for influenza (Goeyvaerts 2015; Vynnycky 2008). In northern latitudes, it can be assumed that the transmission rate is higher in the winter months than in summer (Kronfeld-Schor 2021). This leads to increased infection of the population in the colder seasons, which in turn leads to a temporarily increased immunization of the population, which then decreases again and again.
Fatal Coincidence: Loss of Immunity and Seasonality
While in the winter months a dynamic infection process leads to extensive immunization of the population (strong “infection”), the opposite occurs in the warmer season: the decreasing transmission rate leads to fewer infections and the loss of immunity that occurs at the same time predominates in the population. The increasing number of susceptibles forms the basis for another severe epidemic in the autumn and winter months. The infection thus becomes endemic, with a tendency to recurrent epidemics.
This article describes and discusses the consequences of this coincidence of loss of immunity and seasonally fluctuating transmission rates. Some of the conclusions are trivial, such as that the benefit of a vaccination campaign carried out in the spring, for example, is questionable if the protective immunity of a vaccination only lasts four to six months. In this case, the natural loss of immunity in the population cannot be offset by vaccination. On the other hand, the consequences can be complex and unexpected. This is shown in connection with a characteristic figure for epidemics, the so-called basic reproduction number R0 (Dietz 1993).
It turns out that the direct proportionality between R0 and the peak height of the epidemic, known for the SIR model, is reversed: a highly infectious disease (high R0) can generate high outbreaks in the SEIRS model, which are unexpected with low peak values appear harmless. Infectious pathogens that we consider to be more harmless due to a low R0, on the other hand, appear unexpectedly more strongly and with higher peak values in the SEIRS model. This publication is not intended to address the complex connection between infection and disease, but it can be assumed that more severe outbreaks of infection generally lead to a higher peak load on the healthcare system.
The two sides of the endemic coin
These considerations show that some of the common epidemiological expectations will need to be rethought if SARS-CoV-2 becomes endemic under the coincidence of loss of immunity and seasonal transmission rate. It is also evident that the term “infestation”, which has predominantly negative connotations, is given a second perspective, because infections simultaneously lead to immunization of the population. The consequence of this is that prevented infection of the population also means prevented immunization - the second side of the coin.
There are indications that a new challenge will have to be overcome when planning prevention and intervention measures under Corona: How can health policy measures find the optimal balance between “damage caused by uncontrolled infection” and “protection through immunization”? Vaccination campaigns do not necessarily only have positive and protective effects, but can also have negative epidemiological consequences at the population level. Likewise, measures to reduce contact that are not optimally timed can cause more loss of immunity than they contain the infection process. The implications for intervention decisions are discussed in this work.
methods
For our investigations, we use an extension of the basic SIR model, which is the basis for most infection transmission models. As in the basic model, we assume that people are susceptible to infections at birth (susceptible: “S”) and can become infected. The duration of the infection is divided into two phases, the latent phase, in which the infected cannot yet infect anyone (“E”), and the subsequent infectious phase (“I”); The infection is then followed by immunity (“R”). In contrast to the basic model, immunity after a corona infection is only temporary and can be lost again (return to status “S”). If immune people come into contact with an infectious person, this can also “refresh” and extend their immunity (immunity booster status “B”).
The model is extended by vaccinations, which is why three strata are distinguished:
1) children who are too young to be vaccinated, and
2) older children, adolescents and adults who have chosen to be vaccinated, or
3) who have decided against vaccination. Vaccination "willing" are repeatedly vaccinated, but vaccine immunity is lost over time, as well as the natural immunity after infection.
Those “willing” to be vaccinated can also become infected if, for example, they B. come into contact with an infected person after losing sufficient immunity and before the next vaccination. However, your immunity can also be boosted through infection, not just vaccination. The model is shown in Fig. 2 and is described in its mathematical form with the parameter values used here in the appendix.
Results
Unexpected reversal of effect
It is an invariant property of SIR models (which do not take immunity loss into account) that the peak value of an epidemic is proportional to the base reproduction number (R0) of the virus: the higher R0, the higher the peak value of the epidemic is, and the faster it is reached. Conversely, the smaller R0, the flatter and broader the epidemic is and the lower the peak.
This law is reversed in a SEIRS model with loss of immunity and seasonal transmission rate (Fig. 3; note: in the case of seasonally fluctuating transmission, R0 is the annual mean of the time-varying R0 value.) The example of the dependence of the infection dynamics shown in Fig. 3 from the basic reproduction number R0 shows that conclusions from SIR models are no longer easily applicable when there is loss of immunity and a seasonal transmission rate.
During the Corona waves of the past two years, the aim was to avoid overloading hospital and intensive care capacities, which would cause excess mortality due to the undersupply of the population. In this context, the peak value of the epidemic waves was particularly relevant because an overload is to be expected close to the peak of the epidemic. The inversely proportional relationship between the peak value and the basic number of reproductions shown in Fig. 3 appears paradoxical at first glance. At second glance, however, it is obvious and only corresponds to what can be expected under loss of immunity and seasonal transmission rates (see below).
The apparent contradiction to the results of SIR models has a simple reason: a high infection rate leads to a persistently higher level of basic immunization of the population, and this prevents the later development of fulminant epidemics with high amplitude. Conversely, it should be formulated accordingly: Fulminant epidemics with higher peak values are to be expected from those virus variants that we would rather consider to be harmless - measured by R0.
Fig. 3 shows that for pathogens with loss of immunity and a seasonal transmission rate, the peak load on the health system is greater for those virus variants that have a small basic reproduction number, while virus variants with a large R0 maintain a high basic immunization of the population, so that there are no high peak loads on the health system comes.
The basic reproduction number R0 should not be confused with the effective reproduction number Re, which is often reported as the "R value". Values for Re are smaller than R0 and are often close to 1.0 because they take into account all current infection-reducing factors, while R0 reflects, in a sense, the maximum spread potential of an infection. Since all spread-reducing and time-varying factors such as immunity and contact restrictions are taken into account separately in the model, various infectious pathogens can be better characterized by R0.

Limited effects of vaccination campaigns
Effective vaccination campaigns require a sufficiently high vaccination rate and effective vaccines. In the case of the Corona vaccination campaigns in Germany, both are only available to a limited extent. The vaccination rate in Germany is currently around 75% (RKI vaccination rate monitoring) and the effectiveness of the available vaccines - depending on the virus variant and the vaccine considered - spreads over a wide range, roughly estimated between 60% and 90% (Fiolet 2022). The effectiveness of a vaccination campaign is determined by the product of both percentages, which quickly brings the overall effectiveness of the vaccination campaign into the range of 50% or even lower (e.g. 0.75*0.70=0.52 = 52%).
If a virus has a high infection potential (high R0), then a low effectiveness of the vaccination campaign means that the population is immunized predominantly through natural infection and no longer through vaccination, as desired. The virus's ability to infect then compensates for any ineffectiveness of the vaccination campaign: under high R0, 'leftover' susceptibles are infected very quickly and immunized through infection. This relationship is shown for different vaccination rates and vaccine efficacies in Fig. 4. The model used suggests the following statements in this regard:
1. Immunization of the population: The vaccination campaign has little additional influence on the immunity in the population if a highly contagious virus “exploits” or thwarts any form of non-immunity (= susceptibility) through rapid infection. It hardly matters anymore where the susceptibility comes from, because this can have a variety of causes: low vaccination participation, low effectiveness of the vaccines, unvaccinated population groups (children, people who cannot be vaccinated), delayed vaccination, etc. Any existing susceptibility in the population is then converted into (temporary) immunity through infection with the highly contagious virus. Additional immunization through vaccinations is therefore becoming less important.
2. Endemicity: When an infection has a high R0, is transmitted seasonally and has a pronounced loss of immunity, epidemics can no longer be prevented by vaccination; recurrent epidemics are to be expected, with peak levels proportional to the proportion of the susceptible population before the onset. The results shown in Fig. 4 were generated with R0=6, but are qualitatively also valid for smaller and larger values. Even with an effective vaccination campaign (multiplication of vaccinated proportion and vaccination effectiveness), the epidemic waves are hardly noticeably reduced. The reason for this is the continuous loss of immunity in the population, which leads to a constantly existing pool of susceptible people as the starting point for the respective epidemic wave that follows. The more susceptible people there are in the population, the more pronounced a second outbreak will be later in the year. This depends on the assumptions about the duration of immunity; with very short-lived immunity, the model predicts smaller outbreaks in spring or even summer.

Effects of vaccinations
The studies shown here do not look at the impact of interventions on the disease process, but only the infection process. To the extent that immunizations prevent disease, they benefit both individuals and the health system. But even if the benefit of a vaccination for the vaccinated person is greater than their individual vaccination risk, the vaccination can have epidemiological side effects that are harmful in the population average. This problem has been known since the 1980s (Dietz 1985) and was first documented during a vaccination campaign against rubella (Panagiotopoulos 1999).
An undesirable side effect
A vaccination campaign against Covid-19 occurs when the continuous loss of immunity in the population with the vaccination campaign becomes greater than it would be due to unhindered endemic immunization (natural infection). This constellation is to be expected if a limitedly effective vaccination campaign increases the level of susceptibility of the population and does not reduce it, as intended.
Figure 4 shows such a constellation in scenario C: A highly effective vaccination campaign means that the extent of the winter epidemic is slightly larger if vaccination is used. This is particularly the case
1. if booster vaccinations sometimes only take place after a significant loss of immunity has already occurred, i.e. “too late”;
2. When the immunity generated by vaccination is shorter-lived than after natural infection (vaccination makes individuals susceptible again at an earlier point in time); and
3. if vaccination intervals are in an unfavorable relationship to the loss of immunity after vaccination (in extreme cases, vaccination in the spring after 6 months of immunity leads to new susceptibility in autumn and winter, precisely when a new epidemic develops).

Figure 4 can also be interpreted in two ways:
1) Significantly different vaccination scenarios do not necessarily lead to significant changes in the infection process.
2) Different vaccination scenarios can produce an unexpected effect reversal, such that a desired intervention benefit turns into the opposite, namely an unexpected increase in the peak value of the seasonal epidemic.

Under these circumstances, the net effects resulting from various parameter assumptions (virus variant, effectiveness of the vaccination campaign, duration of immunity after infection and vaccination, R0 etc.) can hardly be predicted without a model.
This generally leads to the question of the extent to which the benefits of a vaccination campaign can be predicted if there are large uncertainties in key figures, parameter estimates or in data (see discussion).

Effect of contact restrictions
These considerations can be continued for contact restrictions or, more generally formulated, for all measures that reduce the transmission rate in the population: A reduction in the number of infections (avoiding contamination) necessarily leads to a lower immunization of the population (see above). This means that any period of reduced transmission causes a loss of immunity in the population. Such a loss of immunity must be compensated for at a later date by allowing infections to spread until herd immunity is achieved (Randolph 2020).
As a rule, this means that the infection process accelerates after a reduction in contact, i.e. the incidence increases. Exemplary scenarios are shown in Fig. 5.
discussion
The article presented here discusses the predictions of a mathematical model on the future endemicity of Corona, with special attention to the interaction of immunity loss and seasonal transmission rate. Both factors have been discussed in the scientific literature (Baraniuk 2021; Kronfeld-Schor 2021), but have hardly been taken into account in health policy and epidemiological decisions, not even in the current debate on the introduction of compulsory vaccination.
The model results shown here are limited to infection epidemiological connections and do not make any statements about clinical aspects, the disease process and the disease burden of corona infections. Since the focus in Germany has so far been on counteracting an overload of available hospital capacities, infection epidemiological connections have been pushed into the background. However, the expected endemicity of Corona in the future is predominantly determined by the infection process, while the disease process follows this with a certain proportionality, which depends on the virus variant but also on immunity protection against disease (RKI 2022b).
Normally it is desirable that a vaccination not only reduces the likelihood of illness, but also the transmission of the infection. In the case of SARS-CoV-2 infections, however, reducing infection transmission through previous vaccinations is proving to be more of a problem. In order to find an optimal compromise between these two opposing effects, expanded models should be used to examine whether vaccinations should perhaps be prioritized specifically for particularly vulnerable population groups.
A mathematical model is a tool that translates assumptions into predictable quantities. The model forecasts depend on two fundamental factors:
(1) the construction of the model (see Fig. 2) and the underlying assumptions, and
(2) from the parameter values used (see Methods).

Based on the current state of knowledge, we must assume that both factors are subject to a significant degree of uncertainty.

Regarding (1): There currently seems to be no consensus about which basic building blocks a mathematical model must contain to predict the endemicity of Corona, and the diffuse methodological situation indicates a need for improvement (Müller 2021). The model used here focuses on the effects of immunity loss and seasonal transmission rate. Other models focus on aspects such as heterogeneity of the population, consideration of risk groups, the influence of different virus variants, or similar factors (Stegmaier 2021).
Deterministic models (like the one used here) assume an infinitely large population and a homogeneous mix of contacts within the population. The length of stay in the individual compartments follows an exponential distribution. These and other implicit and explicit assumptions influence the model predictions as well as the model structure shown in Fig. 2 per se.
Regarding (2): The statements made here also depend on the quality of the input data and the assumptions on the model parameters used. An overview of this can be found at the RKI (RKI 2022a). This starts with assumptions about basic reproduction numbers, which vary widely (Alimohamadi 2020; He 2020; Locatelli 2021; Salzberger 2021) and goes on to our limited knowledge of immunity and loss of immunity; In terms of modelling, this is practically limited to the fact that we cannot assume lifelong immunity after surviving a corona infection, while at the same time a “medium” duration of immunity can hardly be quantified (Baraniuk 2021; Randolph 2020; Schiffner 2021). For immunity after a vaccination, there seems to be the (rather politically shaped) agreement that it could last about six months. There are also indications that immunity after surviving an infection could be longer lasting than immunity after vaccination (Schiffner 2021).

Corona viruses bring with them two properties that have not occurred simultaneously in this way in the previous spectrum of virus infections: a high basic reproduction number and at the same time a short-lived immunity after infection and vaccination. Both characteristics favor endemicity with cyclical (recurrent) epidemics. However, the infectious diseases that have occurred so far can mainly be divided into two other groups:
(1) infectious diseases with high R0, leaving a very long immunity (“childhood diseases”), and
(2) Infectious diseases with a small R0, but which produce only short-lived immunity (e.g. "cold diseases").
The obvious question is which strategies can best be used to combat such a new type of virus. Here, one will first have to agree on a basic model structure, and thus on the relevant influencing factors: Are immunity loss and seasonal transmission rate decisive factors?
If so: What does an adequate basic mathematical model look like under this assumption? The next step will be to improve the quality of the input data and parameter estimates. The past two years of the pandemic have shown that not every available date provides usable information; empirical research is needed at such points.
Predictions and risk assessments should not be derived from individual models, but rather from several, independently developed models. The article presented here is therefore intended to stimulate considerations that the same research questions on the endemicity of Corona should be examined independently by several working groups in the future. This is the only way to create a body of evidence that allows us to make good decisions for the future. This currently also affects the debate about the introduction of compulsory vaccination. <<

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