Integrating Genomic, Climatic, and Immunological Factors to Analyze Seasonal Patterns of Influenza Variants

Abstract

Influenza, often referred to as the flu, is an extremely contagious respiratory illness caused by influenza viruses, impacting populations globally with significant health consequences annually. A hallmark of influenza is its seasonal patterns, influenced by a mix of geographic, evolutionary, immunological, and environmental factors. Understanding these seasonal trends is crucial for informing public health decisions, including the planning of vaccination campaigns and their formulation. In our study, we introduce a genotype-structured infectious disease model for influenza transmission, immunity, and evolution. In this model, the population of infected individuals is structured according to the virus they harbor. It considers a symmetrical fitness landscape where the influenza A and B variants are considered. The model incorporates the effects of population immunity, climate, and epidemic heterogeneity, which makes it suitable for investigating influenza seasonal dynamics. We parameterize the model to the genomic surveillance data of flu in the US and use numerical simulations to elucidate the scenarios that result in the alternating or consecutive prevalence of flu variants. We show that the speed of virus evolution determines the alternation and co-circulation patterns of seasonal influenza. Our simulations indicate that slow immune waning reduces how often variants change, while cross-immunity regulates the co-circulation of variants. The framework can be used to predict the composition of future influenza outbreaks and guide the development of cocktail vaccines and antivirals that mitigate influenza in both the short and long term.

1. Introduction

Influenza viruses, which cause the seasonal flu, are divided into subtypes A, B, C, and D, with subtypes A and B being the most significant for human health. Influenza A is known for its ability to evolve and can be further divided based on the proteins on its surface, leading to different subtypes like H1N1 and H3N2 [1]. This makes type A the main driver of widespread flu epidemics and pandemics. In contrast, influenza B is more stable and does not cause pandemics but can still lead to significant outbreaks [2]. It is split into two lineages, B/Yamagata and B/Victoria, and is mostly found in humans. This limited host range restricts its genetic diversity compared to influenza A, but it remains an important part of the seasonal flu.

The alternation and co-circulation patterns of these viruses vary each flu season, with either influenza A or B, and with different subtypes of A predominating in different years and regions [3]. This variability is influenced by factors like viral mutations, population immunity, and climate conditions [4]. These conditions work together to drive new surges each season, which may be dominated by a single virus subtype or characterized by the co-circulation of both influenza A and B strains. Understanding the drivers behind patterns is crucial for predicting flu seasons, designing effective vaccines, and managing public health strategies.

Epidemiological models are essential for understanding and predicting disease spread within populations [5]. These models fall into two broad categories: deterministic and stochastic. Deterministic models, such as the Susceptible–Infectious–Recovered (SIR) model [6,7,8], predict average outcomes using fixed parameters and differential equations. Stochastic models, on the other hand, incorporate randomness and uncertainties inherent in disease transmission [9,10], providing a range of possible outcomes. Additionally, these models can be compartmental, dividing populations into groups like susceptible, exposed, or infectious [11], or agent-based, where people are simulated as individual entities [12,13]. Immuno-epidemiological models integrate the effects of immunity on epidemic progression. Some combine within-host infection and between-host transmission into a single framework [14,15,16]. Other immuno-epidemiological models are continuous and incorporate population immunity into infectious disease dynamics [17,18,19,20].

Variant competition has been previously studied using multi-strain models, which describe the transmission dynamics of co-circulating strains using a set of equations for each strain [21,22,23,24]. These models, compatible with phylogenetic data, can predict competition patterns between strains [25]. They can be analyzed both analytically and numerically but require assumptions about the seeding of new variants. This limits their use in studying variant emergence dynamics. At the host level, competition between variants has been modeled using similar multi-strain approaches [26]. Reaction–diffusion equation-based models have been developed to examine relationships between viral load, virulence, genotype distribution, and immune response strength [27,28]. Similarly, genotype-structured models have been used to study variant emergence at the population level [29]. This model assumes that infection rates depend on the strain infecting the susceptible individual and that virus transmissibility varies according to strain genotype. Genotype-structured models have also been applied in ecology [30] and tumor growth studies [31,32].

Mathematical modeling has also provided key insights into the dynamics of influenza variant competition [33]. A previous study revealed that small, seasonal changes in the influenza transmission rate can cause large oscillations in incidence due to dynamical resonance [34]. Researchers have used models to forecast seasonal influenza outbreaks over several years [35,36]. Other studies developed models to investigate changes in age-specific protective immunity following influenza surges [37]. Additionally, immuno-epidemic models have offered insights into the factors driving the evolution and seasonal dynamics of influenza [38]. A recent phylodynamic analysis identified cross-immunity as a factor that shapes the alternation between influenza A and B [39]. Another study showed that using vaccines with higher cross-immunity could slow the evolution and spread of influenza [40]. In another work, the impact of immune waning was shown to influence the size and timing of seasonal influenza epidemics [41]. Modeling was also applied to understand the impact of antivirals on influenza transmission [42].

To address the increasing complexity of the immunological and evolutionary landscape of COVID-19, we previously developed a new epidemiological modeling framework that tracks population immunity [17]. This model uses logistic functions, fitted to genomic data, to describe the growth of different COVID-19 variants. It computes the average susceptibility and severity of the disease based on past infections and vaccination histories. The model captures immune escape by considering how emerging variants reduce the efficacy of prior immunity and models immune waning as an exponential decay of immunity. Initially, we used this model to make accurate projections of the burdens from surges driven by Omicron variants. Subsequently, it provided timely scenario projections to the COVID-19 and flu Scenario Modeling Hub [43]. More recently, it has been used to project the combined impact of COVID-19, flu, and RSV in “tripledemic” scenario projections [44] and to determine equitable vaccination strategies for protecting the vulnerable Hispanic population in El Paso County, Texas [45].

In this study, we expanded our previously developed immuno-epidemiological modeling framework to include virus evolution and used it to explore the factors that influence the alternation and co-circulation patterns of seasonal influenza variants. It describes the alternation between influenza A and B by considering a symmetrical fitness landscape. We parameterized the model based on the transmission, climate, immunological, and evolutionary characteristics of influenza in the US. We then validated the model using historic surveillance data on the weekly confirmed influenza cases from a pre-COVID-19 season. Next, we examined the effects of immunity breadth and robustness on the co-circulation and alternation of influenza A and B. We conclude this paper by discussing the opportunities and limitations of this new framework.

2. A Genotype-Structured Model of Flu Transmission, Immunity, and Evolution

2.1. Model Description

This section introduces a new epidemiological model structured according to virus genotype. Influenza transmits through respiratory droplets, direct contact, and contaminated surfaces [46]. Virus evolution and host immunity significantly modulate the virus’ transmission dynamics [47,48]. Viral mutations cause antigenic drift [49], allowing the virus to evade existing immunity and cause seasonal outbreaks. Antigenic shift, involving more substantial genetic changes, can lead to epidemics. Host immunity, acquired through previous infections or vaccinations, exerts selective pressure on the virus, shaping its evolution and influencing transmission patterns. Understanding these factors is crucial for predicting influenza outbreaks and developing effective control strategies.

The new model incorporates a state variable to track population immunity, similar to our earlier models for COVID-19, flu, and RSV [17]. These models assume that the number of recovered individuals contribute to the level of population immunity, which, in turn, reduces the rates of disease transmission and mortality. This approach allows the incorporation of immunological dynamics into compartment-based epidemiological models in a flexible way. The virus genotype, denoted by x, is considered within a one-dimensional genotype space X, representing the genetic landscape for virus transmission. We use an arbitrary genetic unit (G.U.) to measure the genotype and define the boundaries of the genotype space X from 0 to 10 G.U. These values are arbitrary and used to quantify changes in the virus genotype following host-level mutations that impact the virus’s genetic landscape at the population level. They serve as a metric to measure the relationship between key parameters such as inter-variant genetic distance and cross-immunity. The model tracks the changes in the population of susceptible (S), infected (I), and recovered (R) individuals. The populations of infected and recovered are stratified according to the genotype of the infecting virus, while the population of susceptible individuals is considered to be independent of the virus genotype. Mutations during virus replication alter the genotype of the virus. We capture this effect through a diffusion term incorporated in the equation for the infected individuals, similarly to a recent work [29]. The model tracks the changes in population immunity, using a specific state variable (M), taken to also be stratified according to the genotype of the immunity antibodies in the population. It increases when there are new recoveries and decreases due to immune waning. The model also considers cross-immunity, where the effectiveness of infection-generated immunity varies according to the genetic distance between the circulating strain and existing antibodies. Population immunity reduces the transmission rate according to a specific efficacy and cross-immunity with respect to the circulating variants. The dynamics of the model are depicted in Figure 1A.

To analyze the competition and alternation between variants A and B, we consider that the transmission rate varies according to a fitness landscape with two variants: influenza A and influenza B (Figure 1B). We consider that each of these two variants has a radius of 1 G.U., meaning that variant A extends from $x_A-1$ to $x_A+1$, where $x_A$ is the center of the influenza A variant. The two variants are assumed to have the same transmission rate, which corresponds to the basic reproduction number of the flu. The framework follows the dynamics of SIRS models, with the population of susceptible individuals taken independent of the virus genotype as follows:

$${\displaystyle \frac{dS}{dt}}\left(t\right)=-S\left(t\right){\int }_X\beta (x,t){\displaystyle \frac{I(x,t)}{1+K_1{\int }_X{\varphi }_x(y,d_0)M(y,t)dy}}dx+\delta {\int }_{X}R(x,t)dx.$$

(1)

Here, the first term on the right-hand side of the equation represents the infection of susceptible individuals. The transmission rate is denoted by $\beta (x,t)$, while $K_1$ represents the efficacy of immunity in reducing susceptibility, and $d_0$ indicates the breadth of cross-immunity, given in G.U. The second term describes the transition of individuals from the recovered compartment back to being susceptible, a process assumed to occur rapidly according to a rate $\delta$, since the global susceptibility is calculated according to the population immunity [17]. Note that the transmission rate is considered to depend on the fitness landscape as shown in Figure 1B as well as the climate as follows:

$$\beta (x,t)=\beta \left(x\right)\left(1+ϵsin\left({\displaystyle \frac{2\pi t}{365}}+\psi \right)\right),$$

where $\beta \left(x\right)$ is determined according to the basic reproduction number and the genotype as shown in Figure 1B. The parameters $ϵ$ and $\varphi$ are tuned according to humidity data, obtained from the National Centers for Environmental Information (NCEI) [50]. The parameter $ϵ$ was set to 0.2 to indicate a maximum change by 20% in the transmissibility of the virus due to seasonal forcing [44]. Next, we describe the population of infected individuals. We consider these individuals to be structured according to the phenotype of the virus that they harbor:

$${\displaystyle \frac{\partial I}{\partial t}}(x,t)=\theta \left(t\right)\Delta I(x,t)+\beta (x,t)S\left(t\right){\displaystyle \frac{I(x,t)}{1+K_1{\int }_X{\varphi }_x(y,d_0)M(y,t)dy}}-\mu I(x,t),$$

(2)

where the first term on the right-hand side of the equation reflects changes in the genotype of infected individuals. It results from mutations acquired during influenza infection. These changes are driven by virus replication in the infected hosts. Assuming that each host experiences variant replication during infection, we model the genotype change as independent diffusion processes in each host. Assuming that these processes are independent of the genotype (x), we can also represent simultaneous diffusion processes in multiple hosts as a collective diffusion process. To introduce the effect of variant plasticity, we consider a time-dependent diffusion coefficient to consider that the total number of mutations is proportional to the new infections:

$$\theta \left(t\right)=\sigma S{\int }_X\beta (x,t){\displaystyle \frac{I(x,t)}{1+K_1{\int }_X{\varphi }_x(y,d_0)M(y,t)dy}}dx.$$

The mutation rate varies with new infections. The genetic drift, represented by the diffusion coefficient $\theta \left(t\right)$, depends on $\sigma$, which indicates the diffusion rate per new infection as well as the number of new infections. The second term on the right-hand side of the equation for infected individuals describes the infection of susceptible cells, which is downregulated by the level of population immunity (M) and its efficacy ($K_1$). The third term represents the recovery of infected individuals with a rate equal to $\mu$. The impact of population immunity is implemented by considering that the reduction in susceptibility depends on the broadness of population immunity ($d_0$), captured through the following term:

$${\int }_X{\varphi }_x(y,d_0)M(y,t)dy=\left\{\begin{array}{c}{\int }_X{\displaystyle \frac{M(y,t)}{2}}\left(1+cos\left({\displaystyle \frac{|y-x|}{d_0}}\right)\right)\mathrm{if}|x-y|\le d_0\hfill \\ 0\mathrm{elsewhere}\hfill \end{array}\right.$$

Then, we consider the following equation for recovered individuals, also considered structured according to the phenotype of the virus by which they were infected:

$${\displaystyle \frac{\partial R}{\partial t}}(x,t)=\mu I(x,t)-\delta R(x,t),$$

(3)

where the first term on the right-hand side of the equation describes the recovery of infected individuals, and the second term describes the transition of recovered individuals to the susceptible compartment. Note that in our model, we consider that recovered individuals enjoy a short period of protection and then become susceptible again [17]. The level of average susceptibility to infections depends on the level of population immunity. We describe the change in the level of population immunity, stratified according to the genotype, as follows:

$${\displaystyle \frac{\partial M}{\partial t}}(x,t)=k{\displaystyle \frac{R(x,t)}{N(1+K{\int }_X{\varphi }_x(y,d_0)M(y,t)dy)}}-\omega M(x,t),$$

(4)

where the first term on the right-hand side of the equation describes the increase in population immunity due to recoveries, and the second term describes the loss of immunity due to waning. k describes the rate of immunity upregulation due to recoveries, while $\omega$ represents the rate of immune waning. We consider a saturation of immunity generation, which depends on the genotype of pre-existing immunity. This saturation describes the relatively reduced production of antibodies by previously infected individuals. Susceptible individuals typically produce a reduced level of antibodies when they become infected by the same strain for a second time [51]. The simulation is initiated by considering 10 infected individuals with a strain corresponding to influenza A, modeled as a normal distribution with a very small variance equal to $0.02$, to account for the relative diversity of strains that comprise the initially inhaled virus dose:

$$I(x,0)={\displaystyle \frac{10}{0.02\sqrt{2\pi }}}exp\left(-{\displaystyle \frac{1}{2}}{\displaystyle \frac{-{(x-x_A)}^2}{0.{02}^2}}\right)$$

(5)

2.2. Numerical Implementation

We parameterized the model according to flu transmission dynamics in the United States (US). The list of default values for the model parameters is provided in

Table 1. Default values for the parameter used in the model. G.U. denotes an arbitrary unit in the genotype space.

Parameter	Value	Calibration Method or Reference
N	$3.2\times {10}^8$	US population
${\mathcal{R}}_0$	2.3	range from 1.8 to 3 [52]
$\delta$	1/10	short duration [17]
$\sigma$	$50\times {10}^{-9}$ (G.U.)2	arbitrary (drift per each new infection)
$\mu$	$1/5$	influenza recovery time is 3–7 days
$d_0$	3 G.U.	arbitrary
$K_1$	18	95% protection against transmission [17]
k	$3.84$	fitted to seroprevalence data [53]
K	10	immunity saturation rate [17]
$\omega$	$1/(30\times 4)$ day−1	half-life time of immunity is 4 months [54]

. The infection rates for influenza A and B were estimated by fixing the basic reproduction number and recovery rate, and then using the basic reproduction formula to derive the variant 0 infection rate:

$${\mathcal{R}}_0={\displaystyle \frac{{\beta }^{0}N}{\mu }},$$

where N is the size of the population.

The equations are discretized using the Euler explicit method, and the diffusion term in Equation (2) is implemented using the second-order central scheme. These schemes, while conditionally stable, are known for their consistency and accuracy. The time and genotype space discretization steps are set to $dt=1$ day⁻¹ and $dx=0.25$ G.U., respectively. A consistency analysis for the effect of the discretization step of the genotype space is provided in Appendix A. The code was implemented in the Python programming language. The CPU time for a simulation of 22 seasons of 365 days is roughly 23 s on a computer with an AMD Threadripper processor and 64 GB of RAM. The code is available upon request to the corresponding author.

3. Results

3.1. Model Calibration and Validation

The model was validated against the data for the US weekly number of confirmed flu cases in the 2018–2019 season [3]. To reproduce the reported data, we increased the basic reproduction number from the default value of 2.3 to 2.36. As a result, the model accurately reproduced the genomic surveillance data, as shown in Figure 2A. The model predicts that the cumulative number of confirmed cases during the season will total 198,724. It forecasts that the number of reported cases will reach its peak on 5 February 2018. To gain further insights into the contributions of each influenza subtype, we calculated the number of new infections generated by influenza A and B as follows:

$$I_{new}^A\left(t\right)={\int }_{x_A-r}^{x_A+r}\beta \left(x\right)S\left(t\right){\int }_X{\displaystyle \frac{I(x,t)}{1+K_1{\int }_X{\varphi }_x(y,d_0)M(y,t)dy}}dx,$$

$$I_{new}^B\left(t\right)={\int }_{x_B-r}^{x_B+r}\beta \left(x\right)S\left(t\right){\int }_X{\displaystyle \frac{I(x,t)}{1+K_1{\int }_X{\varphi }_x(y,d_0)M(y,t)dy}}dx.$$

In our simulations, $x_A$ and $x_B$ are taken equal to 3 and 7 G.U., and we consider r equal to 1 G.U., as shown in Figure 1B. To compare with the genomic surveillance data, we aggregated the number of cases generated each week to compare with the data. The model indicates that during the 2018–2019 season, influenza A accounted for 90.5% of the cases. To examine long-term seasonal trends, we simulated influenza activity over several consecutive seasons. We assumed that each season starts on 1 September of the year and extends for 356 days. We excluded the first ten seasons from our analysis as it often takes a few seasons for influenza B to emerge and for seasonal patterns to stabilize. Our analysis reveals that the timing of influenza B emergence depends on the considered mutation rate. The patterns in subsequent seasons show a cycle of alternation between seasons predominantly affected by influenza A or B (Figure 2B). Additionally, the seasons dominated by either subtype A or B produced approximately equal numbers of infections. Note that we assumed equal transmissibility for both strains in all simulations. In each season, we can still observe some cases generated by the variant that is not dominant. These cases peak approximately three months after the peak of cases generated by the dominating variant.

To better understand the observed trends, we analyzed the underlying dynamics of population immunity. Figure 2C illustrates the level of population immunity generated by infections with influenza A and B. It shows that the surge driven by influenza A can be attributed to the dominant immunity against influenza B and the waning immunity against influenza A, combined with a slight increase in the transmission rate due to climatic factors. Furthermore, the immunity generated after an outbreak decreases by about half by the end of the season. However, this remaining immunity is sufficient to prevent another outbreak caused by the same variants.

3.2. The Timing of Variant B Emergence Determines the Alternation Patterns of Seasonal Influenza

To study the effect of variant B’s emergence timing, we changed the mutation rate ($\sigma$), which quantifies antigenic drift during infections. The simulation begins with several seasons dominated by variant A. Then, variant B emerges and starts co-circulating or alternating with variant A. Increasing the mutation rate accelerates variant B’s emergence. Higher mutation rates lead to faster viral evolution, reducing the days needed for variant B to appear. We ran four simulations with mutation rates of 25, 50, 75, and 100 $\times {10}^{-9}$ G.U.² (Figure 3).

When the mutation rate is low (25 $\times {10}^{-9}$ G.U.²), variant B emerges at the beginning of season 6, coinciding with a variant A outbreak and causing a significant surge. This surge generates enough immunity to reduce the burden in the following season, leading to alternating significant and mild seasons with the co-circulation of variants A and B. Increasing the mutation rate to the default value (50 $\times {10}^{-9}$ G.U.²) causes variant B to emerge in season 5, without a concurrent variant A outbreak, resulting in a season dominated by variant B. This sets a pattern where variants A and B alternate in predominance.

At a higher mutation rate (75 $\times {10}^{-9}$ G.U.²), variant B also emerges at the beginning of season 5, causing a more significant surge. Further increasing the mutation rate to 100 $\times {10}^{-9}$ G.U.² results in variant B emerging in season 4, dominated by variant A, creating a pattern of alternating significant and mild seasons with the co-circulation of variants A and B.

3.3. The Speed of Immune Waning Regulates the Alternation Frequency of Influenza Strains

After validating the model, we applied it to examine the effects of immunological factors on the alternating patterns observed in seasonal influenza. We conducted three simulations with different half-lives for the waning of population immunity: 4, 8, and 12 months, as shown in Figure 4A. Note that in these simulations, we reset the basic reproduction number to the default value of 2.3. We also analyzed seasonal patterns after the emergence of variant B and the stabilization of the observed dynamics. In the simulation with a 4-month half-life for immune waning, we observed an alternation in the predominance of influenza A and B across seasons. When the immune waning half-life increases to 8 months, we observed a suppressed flu season every three seasons. This season is characterized by very few influenza B cases. The other two seasons alternate between influenza A and B. Further increasing the immune waning half-life to 12 months results in the suppression of three seasons every four seasons, with these seasons characterized by fewer influenza A infections. The remaining season had a higher amplitude and primarily consists of influenza B cases.

On average, the total number of cases per season linked to influenza decreases by 53.6% when the half-life of immune waning increases from 4 to 8 months. Further increasing the half-life to 12 months leads to a 69.74% reduction in the average number of cases per season. Figure 4B shows these average case numbers per season, and

Table 2. Default values for the parameter used in the model. G.U. denotes an arbitrary unit in the genotype space.

Value	Mean Number of Cases per Season	Mean Peak Time
4-month half-life	112,733	5 February
8-month half-life	52,303	22 February
12-month half-life	34,110	23 February

provides the detailed values. The number of seasons dominated by influenza B rises with an 8-month half-life for immune waning and falls with a slower 12-month half-life, as depicted in Figure 4C. The peak of the cases is delayed by approximately two weeks when considering slower immune waning, according to Table 2.

3.4. The Broadness of Population Immunity Determines the Co-Circulation Dynamics of Influenza Strains

After studying the effect of immune waning, we analyzed the impact of the cross-immunity broadness ($d_0$). Cross-immunity represents the immune protection that individuals who have been exposed to one strain or subtype of influenza have against other strains or subtypes, and vice versa. This occurs because some of the immune response components, particularly antibodies, recognize and respond to parts of the influenza virus that are similar between different strains. This cross-protection is usually partial and varies in effectiveness, as influenza A and B are quite distinct overall. To evaluate its impact on seasonal influenza dynamics, we considered four simulations where the value of cross-immunity was taken to be equal to $1,2,3$, or 4 G.U. In taking into consideration that the genotypic distance between variants A and B is 4 G.U., these values correspond to ratios of cross-immunity broadness to genetic distances of 0.25, 0.5, 0.75, and 1. For each of these values, we simulated influenza epidemics during 12 seasons following the emergence of influenza B (Figure 5). Our simulations indicate the existence of two patterns of variant co-circulation and alternations, depending on the value of the cross-immunity broadness.

When the cross-immunity broadness $d_0$ is set to value 1 or 3 G.U., which corresponds to 0.25 or 0.75 of the genotypic distance, respectively, we observed a pattern where influenza A and B alternate in dominance each season. In this case, the size of the epidemic is much larger when cross-immunity broadness is reduced. However, when we adjust $d_0$ to 2 or 4 G.U., which represents 0.5 or 1 of the genotypic distance, respectively, the pattern changes, showing the suppression of one season every two seasons. During the unsuppressed season, there is a significant increase in cases, characterized by the co-circulation of both influenza A and B. These results show that the relationship between the cross-immunity broadness and variant co-circulation patterns is complex and may depend on the synchronization between the cross-immunity and the inter-variant genotypic distance.

To better understand how different cross-immunity broadness values affect variant co-circulation patterns, we analyzed the distribution of infected individuals at the peak of outbreaks. We considered situations where a single influenza variant predominates each season ($d_0=3$ G.U.) and where two variants co-circulate ($d_0=4$ G.U.). Figure 6 shows the distribution of infected individuals by genotype during the peak of two consecutive seasons for both scenarios. In the alternation scenario, most individuals are infected with either influenza A or B, with the distribution slightly skewed away from the opposite variant due to cross-immunity effects. In the co-circulation scenario, we see roughly equal numbers of infections by both variants during major surges, while we report much fewer infections by both variants during mild seasons.

4. Discussion

This study extended our previous immuno-epidemiological modeling framework to include the effects of virus evolution and climate, applying it to explore the factors influencing seasonal influenza patterns. We calibrated the model to replicate the dynamics of influenza A and B in the US during a pre-COVID-19 season. Subsequently, we simulated and analyzed several seasons to understand the dynamics of influenza epidemics over consecutive seasons. Our analysis of changes in variant B’s emergence timing and underlying population immunity dynamics explains the alternation patterns observed in influenza epidemics. These evolutionary and immunological dynamics have shown that the seasonal outbreaks were driven by the waning of immunity generated by infections with a specific variant.

We integrated evolutionary and immunological factors into our model to analyze their impact on seasonal influenza epidemics. Our findings show that the timing of variant B’s emergence determines whether influenza outbreaks follow an alternation or co-circulation pattern. If variant B emerges during seasons dominated by variant A, it leads to co-circulation. Conversely, if variant B emerges when it is absent, the two variants alternate in predominance. After that, we examined how the speed of immune waning affects these patterns by considering different half-life times for the waning of population immunity. Our results show that slower immune waning extends the duration of protection, leading to the complete suppression of influenza outbreaks every few years. Further, we found that longer immunity duration reduces the sizes of epidemics. However, the accumulation of susceptible individuals during one or more mild seasons can result in a disproportionately large outbreak in a subsequent season, which is in agreement with previous findings [41]. We have also shown that the changes in the immune waning speed affect the prevalence of seasons dominated by influenza A or B.

We further investigated the impact of cross-immunity, which regulates the extent to which immunity from one influenza variant provides protection against another. Our simulations show that the broadness of cross-immunity influences the composition of epidemic seasons, altering the dynamics between alternation in predominance by a single variant and the co-circulation of two variants. A previous study identified cross-immunity as a key intrinsic factor in regulating influenza co-circulation patterns, using methods like phylodynamics [39]. Another work, using a multi-strain model, demonstrated the existence of an endemic equilibrium, where strains co-circulate under specific cross-immunity conditions [55].

Our research acknowledges certain limitations. Firstly, we modeled genetic drift in infected populations as a diffusion process, assuming that each mutation’s genetic drift follows a normal distribution, consistent with several previous studies [27,28,29]. This differs from studies using other distributions, such as the log-normal [56]. This approach allowed us to derive a set of reaction–diffusion equations to describe population-level genotype changes. Secondly, our model simplifies the fitness landscape to two variants separated by genetic distance, which may not capture the full complexity of the influenza evolutionary landscape, including the different lineages forming influenza A and B. This simplification is due to our focus on the competition and co-circulation dynamics of influenza A and B, given the availability of data for these two strains. The model can be extended by considering a two-dimensional genotypic space to include the actual antigenic maps for influenza [57]. Finally, our current analysis focused on intrinsic factors driving influenza and does not account for the external effects such as imported cases and vaccination on seasonal influenza epidemics. These extrinsic factors can perturb and alter the steady-state seasonal dynamics.

5. Conclusions

Overall, our study shows that patterns of variant alternation and co-circulation in influenza depend on the synchronization of variant emergence timing, the symmetry of variants A and B, the climate forcing, the rapid immune waning, and the inter-variant cross-immunity. A key characteristic of our model is the incorporation of a symmetrical transmissibility landscape for influenza A and B. This symmetrical landscape is motivated by the partial cross-immunity between the two variants. As a result, it enabled the observed seasonal co-circulation and alternation patterns. This alternation between the variants is driven by selective pressure from immunity build-up and seasonal forcing exerted by humidity, which creates a resonance effect that regulates the timing of these alternations [34]. Additional factors, such as mutation rate and immunity waning rate, further influence the co-circulation patterns and the frequency of variant alternation. These results underscore the importance of using a comprehensive approach that integrates evolutionary, immunological, epidemiological, and climate factors to better understand the seasonal patterns of influenza and design pharmaceutical and non-pharmaceutical interventions that effectively mitigate them. Our flexible framework incorporates these factors, making it suitable for studying the influence of population immunity, virus evolution, and climate change on seasonal influenza dynamics. In the future, we plan to use our framework to predict the variant composition of upcoming seasons and design cocktail vaccines that effectively curb influenza spread based on the dynamics of previous seasons.