1.1 Background

The common cold is a frequent disease in humans and is generally caused by viral infection of the upper respiratory tract [1]. Although mild in most cases, it poses a significant disease burden on individuals and societies, both in terms of human suffering and economic loss. The common cold is the most frequent illness in the US with approximately 25 million documented cases per year [2]. It is estimated that in the US alone, the economic cost of the common cold is approximately $25 billion per year [3]. Despite the large number of cases and the great associated disease and economic burden the common cold is currently not a prioritized research topic. A thorough and precise understanding of the disease dynamics both on a societal and an individual level are lacking today, but might pave the way to better prevention and treatment strategies.

The most frequent causative viral agents are rhino viruses (approx. 30% to 50%), corona viruses (approx. 10% to 15%, not including SARS-CoV-1, SARS-CoV-2 and MERS), influenza viruses (approx. 5% to 15%), respiratory syncytial viruses (approx. 5%), parainfluenza viruses (approx. 5%), adeno viruses (less than 5%), entero viruses (less than 5%), and further unknown viruses (approx. 20% to 30%) [4]. Of these viruses, different strains are circulating and they are constantly subjected to genetic shift and drift.

The immune system is generally capable of clearing a common cold without additional treatment. In healthy individuals the symptoms are often relatively mild (including sneezing, stuffy nose, runny nose, sore throat, coughing, post-nasal drip, watery eyes, fever). Most often, the intensity of symptoms peaks around day 3 or 4 and around day 7 recovery begins [4]. The median duration of symptoms of a common cold has been estimated to be approximately 11 days [5].

In fighting a common cold, different components of the immune system are involved [6]. Generally, the infectious agent enters via the mucous membranes. Here, both specific and unspecific components of the immune system can often already eliminate the infectious agent. In this case, the exposure may result in an asymptomatic course, however, possibly involving training of the immune system. If the infectious agent settles and proliferates, typical symptoms of a common cold may develop. Over time, more powerful components of the specific immune system come into play. In particular, the specific immune system continuously improves its ability to recognize the pathogen and efficiently eliminates it.

After the infection has subsided, the immune system usually retains the ability to recognize the respective pathogen for a while. Hence, after immediate reexposure, it is unlikely that another symptomatic infection occurs. However, with time the newly acquired specific immunity generally deteriorates and may even revert to the baseline level. Furthermore, it is important to note that there is significant cross-reactivity between different virus strains, e.g. in the case of rhinoviruses [7], and possibly even between different viruses. Therefore, a symptomatic infection may be alleviated or prevented after exposure to a virus, if an infection with a similar virus or virus strain has occurred previously (Fig 1).

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Fig 1. Schematic representation of different hypothetical scenarios for immunity upon reinfection with an exemplary virus denoted by “X” on a time scale without units.

In scenario A, upon immediate reinfection with strain “X.1” of the exemplary virus “X”, symptoms are likely to be mild and the infection is usually short or the infection might even be asymptomatic. In scenario B the time between infections is long, and immunity will likely be lost for many viruses and duration of the infection would be long again and might be associated with more severe symptoms. Scenario C considers reinfection with an exemplary strain “X.2” of virus “X”. In this case there might be some cross-reactivity, which alleviates symptoms in case of infection with the related virus strain “X.2”.

https://doi.org/10.1371/journal.pone.0334527.g001

During the COVID-19 pandemic, many regions were subject to long-lasting, extensive contact restrictions, particularly in the winters of 2020/2021 and 2021/2022. Consequently, the incidence of common colds has dropped during this time. The reduced training of immunity to common colds has presumably been linked to a general decline in the population’s immunity to cold viruses [8,9]. In the winter 2022/2023, an increased number of sick days due to common colds was observed in the communities where the contact rates had been lower in the preceding years (e. g. in Germany [10]).

The degree of persistence of specific immunity seems to differ considerably between different common cold pathogens [11]. Infections with rhinoviruses and adenoviruses seem to generally result in long-term, protective immunity against the specific virus serotype, however, not against other serotypes of the virus. Infections with coronaviruses, parainfluenzaviruses, RS-viruses and multiple other common-cold-viruses seem to usually result in short-term immunity, that declines over time.

There is a large corpus of mathematical models describing the dynamics of infectious diseases in a population. Among these models, the classical Susceptible-Infected-Recovered (SIR) epidemic model introduced by [12] holds a prominent place. It is an ordinary differential equation (ODE) model, in which a susceptible fraction of the population (compartment ) can get infected with an infectious disease and hence transition to compartment . Finally, individuals in compartment recover and hence pass into compartment , in which they are immune to the infectious disease. There are numerous variants of this model. One variant that is relevant in the context of this paper is the SIRS model, in which immunity in compartment is lost with a certain rate and individuals hence transition from compartment back into compartment . Another variant of the SIR model is the SIS model, described e. g. by [13,14], in which no immunity is acquired (no compartment ) and individuals pass directly into compartment when recovering.

In the classical SIRS model, the loss of immunity occurs instantaneously after transitioning from to and is hence a binary (immune / not-immune) rather than a gradual property. Assuming a constant transition rate, the resulting duration of immunity is exponentially distributed in a probabilistic setting. Heffernnan and Keeling [15] have examined modifications to this model, examining different waning schemes where immunity is lost gradually rather than abruptly. Specifically they studied stochastic SIR-like models. They conclude, that faster waning of immunity leads to a higher disease prevalence in the population. Khalifi and Britton [16] studied further variants of immunity waning within deterministic ODE, in particular also SIRS-like, models. In a subsequent study, Khalifi and Britton [17] explored optimizing vaccine schemes in the context of immunity waning in SIR-like models and provided recommendations for effectively achieving herd immunity.

The common cold syndrome may be caused by a multitude of different cross-reacting pathogens, potentially resulting in virus-specific and for some viruses cross-virus immunity. These complex immunological interactions between different viruses have been studied using several mathematical SIR-type models (e. g. [18], [19], [20]).

Waning of immunity following SARS-CoV-2 infection or vaccination has been studied in detail, providing quantitative estimates for the temporal decline in immune protection [21]. Recent research has further examined how the frequency of reinfection depends on the rate of immune waning, offering insights into long-term epidemiological dynamics [22]. However, these studies have primarily focused on incidence patterns without explicitly assessing the resulting overall disease burden.

A modified SEIR model incorporating explicit transmission dynamics, waning immunity, behavioral responses, and vaccination strategies has also been developed, demonstrating that long-term social confinement, reduced contact rates, and moderately effective vaccination programs can significantly reduce infection peaks [23]. Yet, also in this study, the net disease burden was not explicitly quantified.

In a pathogen-agnostic approach, it has been analyzed how immunity acquisition and waning of immunity impact on expected death rates [24]. Even there, the net disease burden of non-fatal courses has not been analyzed.

1.2 Objectives

The relation between disease burden per capita per time (referred to as mean disease burden from hereon) and the exposure frequency to particular viruses typically causative of common colds has not been explicitly studied. Exposure frequency may depend among other aspects on overall interpersonal contact patterns and contact reduction strategies of infected individuals. In a certain range, increased exposure likely increases the spread of viral disease and, thus, increases the mean disease burden for most viruses causing common colds. However, we want to investigate if this relation might reverse under certain circumstances, if a critical exposure frequency is exceeded due to increased training of the immune system. In other words, more contacts and hence more exposure to pathogen might strengthen the immune system so that for some pathogens the mean disease burden related to this virus might be reduced. Even though this seems plausible, there are neither reliable epidemiological data available demonstrating such an effect for the common cold, nor has this aspect been studied explicitly in a mathematical modeling framework to our knowledge.

Therefore, we aim to investigate this hypothesis by applying a mathematical modeling approach. Mathematical models express hypotheses in formal, quantitative terms and can be used to evaluate the implications of different hypotheses and design informative experiments. In the following, we present two novel, related mathematical models in order to address the aforementioned research question.

In our analysis, we consider an exemplary virus that can cause an upper respiratory infection. Although the common cold can be caused by a wide variety of viruses, the clinical manifestations—particularly the duration and characteristics of symptoms—often show substantial overlap. Our modeling approach is designed to capture these shared features, allowing us to describe the typical behavior of common cold infections within a generalized framework. However, we acknowledge that the underlying immunological responses can differ significantly across pathogens, particularly with respect to the development and duration of pathogen-specific immunity. To maintain conceptual clarity and analytical tractability, we deliberately neglect immunological interactions between different pathogens. This simplification allows us to focus on core mechanisms while preserving flexibility: the biological properties of the modeled pathogen, such as infectivity and recovery dynamics, can be adjusted within the framework to represent a range of plausible scenarios.

The proposed methods can be applied to any real common cold pathogen by choosing appropriate parameter configurations or estimating them from respective data. Our goal is to find a simple and universal mathematical framework that describes the progression of a typical common cold rather than the precise analysis of a specific pathogen.

Our work is based on the prototypical SIR model family, which allows for an easy connection and integration into the current scientific discourse. The first model that we call CC-ODE model (common cold ordinary differential equation model) is an ODE model based on the SIRS model. In contrast to the plain SIR model, the SIRS model allows to describe loss of immunity. A formal description of the CC-ODE-model can be found in Sect 2.1 and analysis results in Sect 3.1. The second model is an individual-based model derived from the SIS model and is referred to as CC-IB model (common cold individual-based model). Individual-based models allow to follow individuals and their properties [25]. In the CC-IB model, immunity is represented as an individual-specific state. Hence, the explicit modeling of the compartment is not necessary and the SIS model is sufficient as a basis. The CC-IB model is described in Sect 2.2 and Sect 3.2.

The two chosen modeling approaches shall complement each other, the CC-ODE model being better suited for deriving analytical results in closed form and deriving population-centered results, while the CC-IB model inherently allows to represent heterogeneity among individuals, stochasticity and tracking of individual fates.

2.1 CC-ODE model

In this section, the CC-ODE model (common cold ordinary differential equation model) is presented and described. It is an ODE model based on the SIRS model, a classical model describing the dynamics of an infectious disease on a population level. In the SIRS model, individuals can switch between three different compartments. First, the susceptible individuals are in compartment . They do not carry the disease and can potentially get infected. The number of individuals in this compartment at time t is given by . Second, the infected individuals are in compartment and can spread the disease. The number of individuals in this compartment at time t is denoted by . Third, the recovered (and immune) individuals are in compartment . They can neither acquire nor spread the disease and the number of individuals at time t reads . The total number of individuals  +  I  +  R is fixed (phenomena such as birth, death and migration are not included in the model). For reasons of simplicity, we assume N = 1 so that S, I and R can be interpreted as proportions of a population that is constant in size. The model equations read

with initial conditions , and . The model parameters are

• : infection rate (influenced by contact rate and infectivity of the pathogen),
• : recovery rate,
• : immunity loss rate.

Note that by setting the immunity loss we obtain the basic SIR model. This model describes the dynamics of a population exposed to one single virus without interaction to other pathogens or concurrent virus strains. An infection with this virus (strain) can lead to immunity that eventually subsides, which holds true for most common cold pathogens.

The basic reproduction number of this model is . The model implies a steady state solution in which the fraction of infected individuals persists at a constant level over time (endemic solution / equilibrium), which reads:

To investigate if a higher contact rate can eventually lead to a reduced mean disease burden, we introduce some amendments of the equations and parameters leading to an ODE model, which we call CC-ODE model. We assume that an increase of the contact rate leads to shorter (or less intense) infections due to development of specific immunity. Hence, the higher the contact rate, the higher also the recovery rate. However, there is no evidence that the infectivity of the pathogen should directly affect the recovery rate. This is why we split the infection rate β into two independent factors that describe the infection rate by a contact rate () and a measure of infectivity (), which can be interpreted as the probability of infection upon exposure:

In order to represent the possibility of development of specific immunity to the pathogen (= habituation effect), we introduce the parameter α representing the immunogenicity of the virus, giving rise to an additional term in the model equations:

with the additional model parameters

• : infectivity,
• : contact rate,
• : immunogenicity, i. e. a measure of the degree of development of specific immunity to the pathogen in the population upon exposure.

measures the infectivity of the pathogen. Large values of correspond to highly contagious diseases. The contact rate describes the number of contacts per individual in society, with high values indicating a very active population with many contacts. In this way, the higher both of these parameters, the higher the total infection rate. If the infectivity (non-contagious disease) or contact rate (absolute isolation), there is no spread of the disease. The parameter α describes the immunogenicity of the virus, i. e. the immunological habituation effect of the specific immune system against the virus in question. For , there is no development of specific immunity at all and we obtain the classical SIRS model.

For immunogenicity and contact rate , development of specific immunity is so effective that infections are eliminated immediately.

In the current version of the model, immune system boosting depends solely on the contact rate (), not on (the probability of infection per contact). This reflects the assumption that immune stimulation may occur even in the absence of full infection and significant viral replication—i.e., that exposure alone can be sufficient to trigger an immune response.

This model is applied to describe the dynamics of the common cold on a population level. We aim to investigate the disease dynamics in the CC-ODE model by introducing an infection into a small proportion of a population without prior immunity. Thus, the initial conditions read , and for all our simulations of the CC-ODE model.

2.2 CC-IB model

The CC-IB model (common cold individual-based model) is a novel agent-based model derived from the SIS model. The number of individuals is denoted to is an integer > 1 in contrast to the previous section where N was fixed to 1). The dynamics of the SIS model are described by the following equations:

As in the SIR model, the parameter β determines infection rate (composed of number of contacts per person per time and the contagion infectivity), while the parameter γ determines the recovery rate.

The CC-IB model is derived from this model by assuming a population of N individuals, identified by index (). The individuals can switch between and according to probabilistic rules. Each individual can have an individual value for the infection rate β, the value of the i– th individual denoted . Since the infectivity of a virus is an inherent, non-changing property of a specific virus, modulations of directly represent changes in the contact rate in the model. Since infectivity is modeled as constant across individuals, variations in the parameter were interpreted solely as differences in contact rates. The model operates on a discrete time-scale. Each timestep, an individual in acquires an infection (and hence switches to with probability :

A diseased individual currently residing in recovers (and hence switches to ) with a probability that is given by a transition function per timestep, where denotes the time since the last recovery of the i-th individual. is defined as the sum of a term representing the ability of the untrained immune system to clear an infection (c) and a term representing immunity due to previous exposure to the virus. The term representing specific immunity decays exponentially with rate , where d defines the timescale of the immunological memory:

Hence, in summary for each individual identified by index i, the probabilities to switch from one state to the other within the timestep read as follows:

In the simulations, three different parameterizations of are considered that correspond to three scenarios “No specific immunity”, “Medium specific immunity”, “Strong specific immunity” (Fig 2).

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Fig 2. Transition function for the considered scenarios in the CC-IB model.

The transition function describes the recovery probability  +  c (i. e. the probability to switch from the diseased state to the susceptible state ). In the first scenario (“No specific immunity”), no training of the immune system is assumed. In the second scenario (“Medium specific affinity”), it is assumed that an infection results in a mild immunity, which is decaying with time. In the third scenario (“Strong specific immunity”), it is assumed that infection results in a strong immunity, which, however, is also decaying with time.

https://doi.org/10.1371/journal.pone.0334527.g002

The exact values of the parameters a, d, and c in the three scenarios were chosen heuristically in order to produce qualitatively distinct and interpretable regimes of immune response behavior, in particular different recovery dynamics.

The CC-IB model is simulated 10,000 timesteps via Monte Carlo simulations. The population is comprised of 50 “test individuals”, that are representative of the population as a whole since they cover the entire parameter range and 950 additional individuals ensuring a sufficient population size. The trajectories of the test individuals are followed on a per-individual-basis in graphical representations. In the test individuals, ranges from 0.001 to 0.5 equally spaced on a log scale. In the additional individuals, is sampled from a log-normal distribution with μ = –4 and σ = 1.

Initially, without loss of generality, 10 randomly chosen individuals are infected (), the other individuals are susceptible (). It is assumed that none of the individuals have been exposed to the virus before ( for all individuals initially).

A graphical overview of the two presented models and their underlying variants can be found in Fig 3.

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Fig 3. Model overview.

In this figure, the underlying standard SIRS model (upper left corner) and standard SIS model (upper right corner), as well as the novel developed CC-ODE model (lower left corner) and CC-IB model (lower right corner) are shown. The complete model equations can be found in equation 1 (SIRS model), equation 2 (CC-ODE model), equation 3 (SIS model) and equation 4 (CC-IB model).

https://doi.org/10.1371/journal.pone.0334527.g003

3.1 CC-ODE model

The model dynamics for some parameter choices of the immunogenicity α and the contact rate are shown in Fig 4. The parameters α and are of primary interest in this study and are varied systematically in the following. The other parameters are held constant with , and in order to reduce complexity, but could in principle be estimated from biological data, in case it is available for specific viruses.

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Fig 4. Solutions of CC-ODE model.

The solutions are depicted for with initial conditions , , , parameter values , and different choices for immunogenicity and contact rate .

https://doi.org/10.1371/journal.pone.0334527.g004

As one would expect, with an increasing value of immunogenicity α and constant values for all other parameters, there are more susceptible and less infected individuals. The dynamics for varying contact rate appear to be a bit more complex and depend essentially on the choice of α. To further investigate this, we have a closer look on the steady states of the ODE system. There are two steady states of the ODE system. One of them is the trivial steady state , in which there are solely susceptible individuals in , while there are neither infected nor immune individuals in the population. Hence, there is no infection at all that could be spread and the entire population remains susceptible and healthy. The condition

ensures instability of the trivial steady state and hence the basic reproduction number reads

As a consequence, there is a disease outbreak, if . Otherwise, an introduction of the virus to a small part of the population leads to the extinction of the disease. On the one hand, the higher the probability of infection upon exposure and contact rate , the higher the chance that the disease is breaking out. On the other hand, the greater the immunogenicity α of the virus and the recovery rate γ, the lower the probability that the disease is breaking out. This coincides well with the intuitive notion of these model parameters.

Additionally, there is another, non-trivial steady state, representing the endemic equilibrium:

This expression closely resembles the endemic equilibrium of the classical SIRS model with waning of immunity (see Sect 2.1). The above result follows directly from the classical solution by substituting and .

In the following, we focus on this second steady state representing the endemic equilibrium. Thereby, we further investigate and its behavior depending on the parameters α and , see Fig 5 a). can be interpreted as the proportion of infected individuals in the long term and is therefore a suitable representation of the mean disease burden after an initial period. For , we obtain curves with one single maximum at

Hence, for a contact rate , there is a decrease of and thus of the number of infections in the long term. Depending on the specific characteristics of a virus and the resulting different parameters in the ODE model, a higher contact rate of the individuals can lead to an overall lower mean disease burden. That is, having overall more contacts helps to reduce the mean disease burden, if there is sufficient development of specific immunity. Interestingly, with increasing probability of infection upon exposure the location of the maximum is decreasing, see Fig 5 b). This means that for pathogens with higher probability of infection upon exposure, it might be the better strategy to have also a higher contact rate to keep the mean disease burden lower. Note, however, that this may not be reasonable for all diseases. The stability analysis of the steady states can be found in the supplementary material S1 File.

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Fig 5. Analytical results of CC-ODE model.

a) Analytical solutions of steady state component (infected individuals) vs. log-scaled contact rate and . b) Maximum location of steady state vs. pathogen infectivity . The colors code for different choices of the immunogenicity α. Other parameter values are .

https://doi.org/10.1371/journal.pone.0334527.g005

3.2 CC-IB model

The CC-IB model focuses on the dynamics at the individual level. For the CC-IB model, we evaluate three scenarios corresponding to different patterns of immunity development (no specific immunity, medium specific immunity, strong specific immunity). The number of persons in the compartments and over time is depicted in Fig 6. It can be seen that in all three scenarios, the fraction of infected individuals (I/N) oscillates around an equilibrium point.

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Fig 6. Dynamics in the compartments in the CC-IB model.

The system oscillates around an infection level of about 25% in all three scenarios. The solid line is based on locally estimated scatter plot smoothing. The qualitative system dynamics are independent of the starting conditions, provided extinction of the disease does not occur (simulations not shown).

https://doi.org/10.1371/journal.pone.0334527.g006

In order to illustrate the model dynamics, relation between the infection rate and the mean residence time in the two compartments is visualized in Fig 7. For compartment , we can observe that individuals with large values for tend to have shorter residence times. This is to be expected, as exposure to the infectious virus is less likely for small values of . The transition from to is identical for all three scenarios, so that there are no relevant differences between the three scenarios in this regard. In contrast, the mean residence time in displays important differences between the three scenarios. In scenario ‘No specific immunity’, the duration of infections (= residence time in ) shows no dependence of the infection rate. For scenario “Medium specific immunity”, it can be seen that individuals with a higher infection rate (large ) tend to have shorter infections. This relation is even more pronounced in the scenario “Strong specific immunity”.

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Fig 7. Mean residence times in the compartments and .

For all individuals, the mean residence times in and have been calculated and plotted as individual dots on log scale versus the infection rate on log scale. The solid lines are based on locally estimated scatter plot smoothing.

https://doi.org/10.1371/journal.pone.0334527.g007

In Fig 8, we show individual trajectories for the time interval from t = 9000 to t = 10000. It becomes clear that in all three scenarios, individuals with a small are rarely infected and individuals with larger have infections more frequently. In the first scenario (“No specific immunity”), the duration of infections is independent of . In the second scenario (“Medium specific immunity”), infections appear to be shorter for larger values of . In the third scenario (“Strong specific immunity”), this notion becomes more evident. In this scenario, the total time spent in state first increases with increasing infection rate . For large values of , infections are frequent, however, tend to be short in duration.

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Fig 8. Trajectories of test individuals and mean disease burden.

In this figure, time is plotted from bottom to top. Each column corresponds to a simulated test-individual with a particular value for the infection rate . At the times marked dark gray, the individual is in , at the times marked light gray in . The colored dots and the black trend lines indicate the fraction of time spent in with respect to the entire simulation time. The corresponding scale is on the right hand side of the figure.

https://doi.org/10.1371/journal.pone.0334527.g008

The fraction of time spent in in the simulation interval (t from 0 to 10000) is chosen as a proxy for mean disease burden. The fraction of time spent in is the product of the number of infections and the mean residence time divided by the total time. In the scenarios “No specific immunity” and “Medium specific immunity”, there is a monotonous trend to increase with greater (see trend line and colored dots in Fig 8). However, in the scenario “Strong specific immunity”, the mean disease burden tends to decline once the infection rate (and hence the contact rate) surpasses a certain value of . Hence, we confirm in our model analysis that also in populations that are heterogeneous with respect to their contact rates, an increased exposure frequency can lead to a decreased mean disease burden for appropriate parameter configurations, as already shown for the CC-ODE model.