Lightweight Models for Influenza and COVID-19 Prediction in Heterogeneous Populations: A Trade-Off Between Performance and Level of Detail

Abstract

In this work, we employ two modeling approaches—a mean-field model and a network model—for the purpose of modeling respiratory infection outbreaks in Russia. The presented approaches and their software implementation combine heterogeneity and structural simplicity and, in this sense, they close the gap between the compartmental SEIR models and complex detailed solutions based on agent-based approaches—the two most common modeling techniques for influenza and COVID-19 dynamics. The mathematical description of the approaches is presented, with SEIR compartmental model serving as a baseline for comparison. The experiments demonstrate the similarity of the modeling output of the presented approaches, which allows their interchangeable usage in replicating real outbreak dynamics in Russian cities. The ability of the discussed approaches to mimic data from Russian epidemic surveillance is shown by fitting a mean-field model to data from an influenza outbreak in Saint Petersburg in 2014–2015. The comparison of model complexity and their performance is made using synthetic scenarios. Following the results of numerical experiments, the comparative advantages and drawbacks of the approaches in the application to respiratory infection outbreaks are discussed. The presented modeling techniques, in addition to classical SEIR models and agent-based models as a part of epidemic surveillance, allow one to select the best modeling option for any particular task in outbreak surveillance and control, based on the computational resources at hand, data availability, and data quality.

1. Introduction

Influenza remains a significant public health challenge, with profound social and economic repercussions. Seasonal influenza outbreaks not only lead to increased morbidity and mortality, but also impose substantial burdens on healthcare systems and the economy. For example, economic costs associated with influenza include lost productivity due to absenteeism, increased healthcare expenditures, and indirect costs related to reduced workforce efficiency. To improve the preparedness of authorities and medical institutions, rapid and precise reactions are essential. This is where mathematical models come into play, providing a framework for quantifying the dynamics of influenza outbreaks and forming decision-making processes. Mathematical modeling serves as a powerful tool in epidemiology, allowing researchers to simulate and forecast the spread of infectious diseases in various scenarios. Different types of epidemic model exist, each varying in complexity and in the level of detail with which they address the mechanisms of infection transmission. For epidemic surveillance and morbidity forecasting, it is essential to be able to quickly and accurately calibrate the model using known data, as well as to account for the characteristics of contact patterns that influence the spread of disease. Therefore, it is necessary to maintain a balance between the complexity of the model and its ability to be calibrated to real data.

Compartmental models, which are based on ordinary differential equations (ODEs), offer a simplified representation of disease transmission by categorizing populations into compartments—for instance, susceptible, exposed, infected, and recovered [1]. Such models are well studied and can be subject to analytical analysis, but they do not take into account the heterogeneity of the population, nor do they track the spatial distribution of the disease. Multi-agent models can address these issues by examining the process of disease spread at the level of individual contacts. Typical multi-agent models are implemented as algorithms on graphs, where vertices represent individuals and edges simulate connections between people, which may result in infection transmission. Graphs can be constructed based on demographic studies of contact networks or using standard networks, such as Barabási–Albert, Watts–Strogatz, or Erdős–Rényi graphs and so on. The simulation runs for multi-agent models are time-consuming, which is a critical factor when the model has to be launched a number of times with various parameter sets—e.g., in the case of repetitive calibration to data and sensitivity analysis. Network models, which do not contain individual features of population members (e.g., [2,3]), are less demanding in terms of computational resources. Still, the necessity to monitor peer-to-peer infection transmission makes them slower than the compartmental models, which operate with homogeneous compartments without individual connections between the population members. A compromise between the computational complexity and the level of detail in describing an epidemic is represented by models that approximate graph descriptions using differential equations. Such models include mean-field models, pairwise interaction models, and models based on percolation theory.

In this work, we demonstrate two models that combine heterogeneity of regarded population and structural simplicity in the description of the epidemic process. The presented approaches and their software implementation intends to close the gap between the classical compartmental SEIR-type models and complex detailed solutions based on agent–based approaches, which are the most common modeling techniques for modeling influenza and COVID-19 dynamics. Thus we form a set of models with varying structural complexity to be used in epidemic surveillance, which allows us to select the best modeling option regarding the computational resources, data availability, and level of input uncertainty for every particular case related to purposes of epidemic surveillance and control.

2. Methods

In this section, we provide a brief description of the models employed: the compartmental model, the network model, and the heterogeneous mean-field model. Our foremost interest is to investigate the options of using them interchangeably for a given outbreak dynamics simulation. We base our ideas on the works about aligning differential and multi-agent models, e.g., [4,5], which show that it is possible to establish a certain relation between the output of these types of models despite the dramatic differences in their structure. For the idea of how to match the named models, we largely relied on the results provided in book [6]. Since the authors performed all the analysis on the SIR models (the incubation period of the disease is not considered), we extend some of their results for SEIR models (where E stands for exposed, i.e., infected in latent state) and show the applicability of their concept to these models as well. Our necessity to operate with SEIR rather than SIR is based on the fact that the epidemic respiratory infections of our interest, such as influenza and COVID-19, have pronounced periods of latent infection, which makes the usage of SEIR more correct. Namely, the influenza incubation period is known to be around 1–4 days (2 on average) [7], whereas COVID-19 incubation period is on average 5–6 days, but can be as long as 14 days [8] (although it becomes shorter for more recent strains).

2.1. Models

2.1.1. SEIR Compartmental Model

Our baseline model for comparison is a compartmental SEIR model. The compartmental SEIR model assumes the division of the population into four compartments: S—susceptible, E—latent, I—infectious, and R—recovered. The transitions between compartments are described by a system of differential equations:

$$\begin{array}{c}\dot{S}=-\tau SI,\\ \dot{E}=\tau SI-\alpha E,\\ \dot{I}=\alpha E-\gamma I,\\ \dot{R}=\gamma I,\end{array}$$

(1)

$$\begin{array}{c}S\left(0\right)=S_0,E\left(0\right)=E_0,I\left(0\right)=I_0,R\left(0\right)=R_0,\end{array}$$

(2)

where $\tau$ is infection transmission rate, $1/\alpha$ is mean latent period for a newly infected person, and $1/\gamma$ is mean infectious period. Since we regard a single disease outbreak lasting not more than 180 days, we do not consider demographic effects, such as birth, death, and migration; thus, there are no inward and outward flows in the equations for the compartments. We do not distinguish asymptomatic infectious individuals—we take into account their possible existence at the stage of calibrating models to data, together with the incidence cases missed by epidemic surveillance and hence not reported. Also, for the sake of simplicity, we regard only one generic virus strain, although, in the case of influenza, parallel epidemics caused by several strains are common [9].

Compartmental models are well-studied and suitable for analytical analysis, such as the assessment of parameter values that prevent an outbreak [10,11]. The drawback of this approach is that the heterogeneity of the contact network is not taken into account. This means that for all the cities with the same population size, the form of the infection curves will be identical, despite the differences in demographic characteristics and varying population densities in these cities. Another drawback of this approach is the inability to track the spatial spread of the disease, as the model’s output consists solely of the compartment sizes over time.

2.1.2. SEIR Network Model

The SEIR network model takes a graph of social interactions as an input. We use network models commonly associated with social connections in the population, which are the Barabási–Albert model [12] and Watts–Strogatz model [13,14]. The time complexity of the algorithm is $O(t_{max}·N·k_{max})$, where $t_{max}$ is the number of simulation steps, N is the number of nodes, and $k_{max}$ is the maximum node degree in the input network. The SEIR network model in this work is implemented based on the principles described in book [6] and in the code for this book, available at [15]. To simulate epidemics on networks, the Gillespie algorithm is used [16].

2.1.3. SEIR Heterogeneous Mean-Field Model

The mean-field model approximates the network properties using a system of differential equations for nodes with a specific degree of vertices k, which allows it to combine the advantages of the two previous model approaches. The derivation of formulas for this model in the case of three compartments (S, I, and R) is described in [6]. Below, we evaluate the equations for the extension of this model, which includes four compartments (S, E, I, and R).

If $X_i\left(t\right)$ is the variable that returns the status of node i at time t, we can define the expected value as follows:

$$[A]\left(t\right)=\sum _{i=1}^{N}P(X_i\left(t\right)=A),A\in \{S,E,I,R\}.$$

(3)

The expected number of edges which connect nodes in a given status A and B can be defined as follows:

$$[AB]{\displaystyle (}t{\displaystyle )}=\sum _{i=1}^N\sum _{j=1}^N{g}_{ij}P{\displaystyle (}{X}_i\left(t\right)=A,X_j\left(t\right)=B{\displaystyle )},A,B\in \{S,E,I,R\},$$

(4)

where $g_{ij}$ is the adjacency matrix value.

Firstly, we consider a pairwise model:

$$\begin{array}{cc}\hfill [\dot{S_k}]& =-\tau [S_{k}I],\hfill \\ \hfill [\dot{E_k}]& =\tau [S_{k}I]-\alpha [E_k],\hfill \\ \hfill [\dot{I_k}]& =\alpha [E_k]-\gamma [I_k],\hfill \\ \hfill [\dot{R_k}]& =\gamma [I_k],\hfill \\ \hfill k& =1,2,\dots ,M,\hfill \end{array}$$

(5)

where k is the degree of a node and M is the maximum degree among the nodes in a network G. We can perform mean-field approximation using a multiplier ${\pi }_I$—the probability that a random stub in a network is connected to a stub of an infected node:

$${\pi }_I={\displaystyle \frac{{\displaystyle \sum _{l=1}^M}l·[I_l]}{{\displaystyle \sum _{l=1}^M}l·N_l}},$$

(6)

where $N_l$ denotes the number of vertices with degree l in graph G. Therefore, the average number of edges linking susceptible nodes with degree k to infected nodes can be approximated as

$$[S_{k}I]\approx k[S_k]{\pi }_I.$$

(7)

Now, we can rewrite equation for $[{\dot{S}}_k]$:

$$[\dot{S_k}]+\tau k{\pi }_I[S_k]=0.$$

(8)

Thus, $[S_k]$ can be expressed as follows:

$$\begin{array}{cc}\hfill [S_k]\left(t\right)& =S_k\left(0\right)·{\theta }^k\left(t\right),\hfill \\ \hfill \theta \left(t\right)& =e^{-\tau \underset{0}{\overset{t}{\int }}{\pi }_I\left(\widehat{t}\right)d\widehat{t}}.\hfill \end{array}$$

(9)

Finally, we can formulate a system of equations for the SEIR heterogeneous mean-field model:

$$\begin{array}{cc}[S_k]\hfill & =S_k\left(0\right)·{\theta }^k\left(t\right),\hfill \\ \hfill [\dot{E_k}]& =-[\dot{S_k}]-\alpha [E_k],\hfill \\ \hfill [I_k]& =[N_k]-[S_k]-[E_k]-[R_k],\hfill \\ \hfill [\dot{R_k}]& =\gamma [I_k],\hfill \\ \hfill \dot{\theta }& =-\tau {\pi }_I\theta ,\hfill \\ \hfill \theta \left(0\right)& =1,[R_k]\left(0\right)=0,\hfill \\ \hfill k& =1,2,\dots ,M.\hfill \end{array}$$

(10)

To generate the epidemic dynamics, this system is solved numerically, with the corresponding code for the simulation implemented in Python language.

2.2. Performance Metrics

To assess the suitability of models for the use in epidemic surveillance, we calculate the following model metrics:

• Average time ($T_{avg}$) of one simulation run;
• The number of free parameters k. Larger k indicates a higher structural complexity of the model. This value was also used by us in assessing model calibration accuracy with regard to their complexity [17] based on Akaike criterion [18] and the corresponding AIC indicator [19];
• Number of equations (when applicable).

3. Results

3.1. Simulation Results

For the proposed SEIR heterogeneous mean-field model, a series of numerical experiments were conducted to compare the generated epidemic dynamics with those of the SIR model. The parameters for the models are presented in

Table 1. SEIR compartmental model parameters description.

Parameter	Description
$\tau$	Infection transmission rate
$1/\alpha$	Mean latent period, days
$1/\gamma$	Mean infectious period, days
$t_{max}$	Simulation time, days

and

Table 2. SEIR network and mean-field model parameters description.

Parameter	Description
$\tau$	Infection transmission rate
$1/\alpha$	Mean latent period, days
$1/\gamma$	Mean infectious period, days
N	Network size (number of nodes)
m	Number of edges to attach from a new node to existing nodes (for Barbasi–Albert network)
k	Number of nearest neighbours to join in ring topology (for Watts–Strogatz network)
p	The probability of edge rewiring (for Watts–Strogatz network)
$t_{max}$	Simulation time, days

.

Figure 1 demonstrates the comparison of newly infected cases (incidence) for SIR and SEIR network models, as well as for the mean-field models of the same population structure. Figure 2 shows a similar comparison for the cumulative number of newly infected individuals (cumulative incidence). Simulations for all models were conducted using the same parameters. The changes in the dynamics of epidemic indicators for heterogeneous mean-field models correspond to similar changes for network models when transitioning from the SIR to the SEIR type. A similar change in the shape of disease incidence curves may be observed when transitioning from the SIR to the SEIR compartments in the case of basic compartmental models. As can be seen in the graph, the mean-field models demonstrate higher incidence compared to network models. This discrepancy is related to the nuances of the mean-field approximation. A similar effect was described for SIR models in [6]. At the same time, it is shown that different types of models (network and mean-field) within one population structure (SIR or SEIR) can be used interchangeably with some parameter adjustment since they produce epidemic curves of similar shape. This is demonstrated in Figure 3, where we present the calibration of the SEIR heterogeneous mean-field model to the output data from the SEIR network model. While the mean-field approximation tends to overestimate incidence rates in comparison to the network model, it can be calibrated to align with the network model’s values and effectively reproduce the epidemic dynamics on networks ($R^2=0.98$).

To demonstrate the ability of the SEIR heterogeneous mean-field model to replicate real epidemic data, we consider the influenza epidemic season of 2014–2015 in Saint Petersburg, Russia. In Figure 4, we demonstrate the results of a parameter sweep of our mean-field SEIR model compared to the original mean-field SIR model from the library [15] with the real epidemic data in the background. As one can see, the modeling curves produced by an SEIR model presented in this paper have a shape more similar to real incidence compared to an SIR model. We might assume nonetheless that the calibration of an SIR model to incidence datasets for Russian cities instead of an SEIR model is theoretically possible, but this would cause problems in the correct interpretation of resulting optimal parameter values. Particularly, by assuming that the modeled disease, either influenza or COVID-19, has no incubation period, we will inevitably add errors to the assessed values of transmission intensity and effective reproduction number $R_t$. The results of SEIR mean-field model calibration to weekly incidence data are shown in Figure 5. The optimal parameters were calculated using manual and automatic calibration. Ultimately, we were able to accurately replicate the incidence curve ($R^2=0.9$).

3.2. Comparison

In this section, the key parameters related to the model performance (average execution time) and the detail of the description of the epidemic process (structural complexity represented by parameter k and the number of equations used to describe the model) were compared for our regarded models. The simulation time for a synthetic outbreak with a peak incidence around 25,000 cases and parameter values related to model complexity are presented in

Table 3. Comparison of models.

Model	Average Simulation Time, s	k	Number of Equations
SEIR-ODE	$4·{10}^{-3}$	4	4
SEIR mean-field	$5·{10}^{-1}$	6 (7) *	$5·L$, where L is the number of distinct degrees in network
SEIR network	${10}^2$	6 (7) *	N/A
multi-agent model	$2·{10}^4$ [20]	22	N/A

* 5 + 1 or 5 + 2, depending on network type: one additional parameter for Barabási–Albert and two additional parameters for Watts–Strogatz network type (see Table 2).

. The compartmental SEIR model consists of only four equations (in Equations (1) and (2)) and has four parameters (see Table 1); it also demonstrates the highest execution speed. The mean-field model and the network model have the same number of parameters k. For the network model, the number of equations is not defined. The mean-field model operates three orders of magnitude times faster than the network model, which is crucial for tasks of epidemic surveillance. The mean-field model has more parameters than the compartmental model because of the additional parameter set related to the input network. The first additional parameter is the size of the network N; the number of other additional parameters vary depending on the selected network type. In the case of a Barabási–Albert network, there is one additional parameter, whereas, for a Watts–Strogatz network, there are two (see Table 2). The number of equations for the mean-field model equals the number of equations required to describe nodes with the same node degree, multiplied by the number of different node degrees L. In our case, the total number of equations amounted to $5·L$ (see Equation (10)). Another model included in the analysis is a detailed multi-agent model that takes a synthetic population of the studied city as input. The description of the model, of the synthetic population, and the method of estimating structural complexity based on the value of k are provided in works [20,21]. The multi-agent model has a flexibly configurable contact network built using demographic and geographic data, which is its strength. On the other hand, its execution time exceeds that of the network model by two orders of magnitude.

3.3. Sampling Approach for Network Models

To reduce the simulation time for the network model, sampling can be employed. To correctly scale from a large population to a smaller one, it is necessary to determine a transformation that allows the conversion of the incidence curve of the small population to the large population case. Particularly, this is a great obstacle for the application of multi-agent models in epidemic surveillance, which our previous studies showed. Figure 6 and Figure 7 show the results for incidence simulation as a function of time using an SEIR heterogeneous mean-field model. In the inset plots, we demonstrate that, due to the properties of regular networks (Barabási–Albert and Watts–Strogatz types), the peak incidence rates and the total number of infected individuals throughout the epidemic increase linearly with population size. Thus, the simulation time using network models can be reduced by decreasing population sizes, and, after the simulation, target epidemic values can be recalculated based on this linear relationship. This is generally not true for the multi-agent models that use synthetic populations because the structure of contacts in such populations may be intricate and cannot be easily sampled.

4. Discussion

In this paper, we examined two modeling approaches that are capable of replicating disease outbreak dynamics in heterogeneous populations: network-based and heterogeneous mean-field SEIR models. Each of them demonstrates distinct advantages and limitations. The proposed SEIR heterogeneous mean-field model combines high computational speed with the ability to account for heterogeneity and other characteristics of the contact network in a population. Thus, this model ideally contributes to the full completion of the range of models, where, on the bottom edge, we place the fastest but the less detailed compartmental model, and, on the top edge, the highly detailed multi-agent model based on a synthetic population of the city. A set of models with varying levels of detail and performance serves as a convenient tool for epidemiological surveillance. Depending on the tasks set by health authorities, different models can be employed that best fit the specific objectives. For example, studying targeted vaccination is not feasible with compartmental models; the most suitable option would be a network model or a multi-agent model, which allow the tracking of the impact of vaccination at the individual level. If it is crucial to consider geographical characteristics of disease spread, such as the density of urban areas, constructing a synthetic population and observing infection dynamics in relation to specific geographical regions becomes necessary with the help of a multi-agent model, no matter the time expenses.

The comparison of models demonstrated the following:

• The network model operates faster than the multi-agent model, allowing for the sampling of contact networks. It is also straightforward to implement since there is no need to construct synthetic populations;
• The heterogeneous mean-field model runs three orders of magnitude faster than the network model, facilitating calibrations against real data. This model can also account for population heterogeneity since it uses the same graph input as the network model. At the same time, the individual-level tracking of infection transmission with this model type is impossible.

It is worth noting that the strengths of various models can be combined within a single framework through a hybrid approach [22,23]. In this approach, at the beginning of an epidemic, either a network or multi-agent model is considered to track disease dynamics at an individual level. As the disease incidence rises significantly, a switch occurs to a more general model, such as the heterogeneous mean-field model or a compartmental model. This strategy aims to jointly enhance detail and reduce simulation time. Additionally, there exists a whole class of models that approximate network models, including pairwise interaction models and percolation models. In future work, these models are planned to be added to enhance smooth transitions in detail and simulation time.