Data-driven deep learning algorithms for time-varying infection rates of COVID-19 and mitigation measures

Epidemiological models with constant parameters may not capture satisfactory infection patterns in the presence of pharmaceutical and non-pharmaceutical mitigation measures during a pandemic, since infectiousness is a function of time. In this paper, an Epidemiology-Informed Neural Network algorithm is introduced to learn the time-varying transmission rate for the COVID-19 pandemic in the presence of various mitigation scenarios. There are asymptomatic infectives, mostly unreported, and the proposed algorithm learns the proportion of the total infective individuals that are asymptomatic infectives. Using cumulative and daily reported cases of the symptomatic infectives, we simulate the impact of non-pharmaceutical mitigation measures such as early detection of infectives, contact tracing, and social distancing on the basic reproduction number. We demonstrate the effectiveness of vaccination on the transmission of COVID-19. The accuracy of the proposed algorithm is demonstrated using error metrics in the data-driven simulation for COVID-19 data of Italy, South Korea, the United Kingdom, and the United States.


Introduction
In December 2019, a new respiratory illness began to spread throughout Wuhan, China. The virus responsible for this illness is the SARS-CoV-2 and the disease is called COVID-19 [1]. It quickly spread through Wuhan, a city of 11 million people in Hubei province. It infected tens of thousands of people over the ensuing weeks. China imposed major restrictions on travel and work, and by the end of February, cases of COVID-19 had slowed inside the country while spiking all over the world. COVID-19 data from different countries reflects various mitigation measures [2,3], such as lockdown, social distancing, early detection of infectives, contact tracing, and vaccination [4,5,6]. Many data-driven approaches in infectious disease modeling are linear models. When using linear regression, statistical methods such as Auto Regressive Moving Average (ARIMA) and Moving Average (MA) rely on assumptions which make it impossible to neural networks are known to be universal approximators of continuous functions [23,24]. Feedforward neural networks (FNN) have been used to learn approximate solutions of differential equations. In [25], FNN was combined with the traditional Cox model for survival analysis to predict the clinical outcome of COVID-19 patients. In [26], FNN was used to develop differential equation solvers and parameter estimators by constraining the residual. This FNN is called the Physics Informed Neural Network (PINN).
PINN has been used to simulate pandemic spread, see [27], where the model parameters were taken to be constants [26,28], PINN was used to solve nonlinear partial differential equations from data. PINN has been used to solve system of ordinary differential equations [29] and system of fractional differential equations [30]. In [31], an algorithm that combines PINN together with LSTM is presented to solve an epidemiological model and identify weekly and daily time-varying parameters.
To overcome the limitations of statistical approaches, we present an Epidemiology-Informed Neural Network (EINN) inspired by applying a PINN to epidemiology models. Given that it may not be possible to know the most accurate form of a time-varying transmission rate, EINN algorithms is a viable option to learn time-varying transmission rate and to detect the impact of mitigation measures from data. The EINN loss function is extended to include some known epidemiology facts about infectious diseases. To detect hidden details in the training data, a cubic spline interpolation is used to generate sufficient training data. The proposed EINN algorithm can capture the dynamics of the spread of the disease and the influence of various mitigation measure. Since asymptomatic infectives population is unreported in the publicly available data [32]. EINN algorithm learns asymptomatic infectives population by training on symptomatic infectives data that are available in the reported public data.
The paper is organized as follows. In Section 2, we introduce and discuss the asymptomatic-SIR model, the neural network structure of EINN and the EINN algorithm for time-varying transmission rate. In Section 3, data-driven simulation results for constant transmission rates, data-driven simulation results for pharmaceutical and non-pharmaceutical mitigation measures, and data-driven simulation results for timevarying transmission rates are presented. In Section 4, we discuss the mitigation measures, vaccination efficacy, the time-varying transmission results and error metrics for data-driven simulation. Finally, a summary of the results in this paper is presented in Section 5.

Asymptomatic-SIR Model
The asymptomatic-SIR model introduced in [16] assumes that some of the infectives are asymptomatic infectives. This group is infectious despite not showing symptoms of COVID-19, probably are not tested, and are usually unreported in the various publicly available data.
The asymptomatic-SIR model considers the following population compartments: the Susceptible (S), the symptomatic Infectives (I) which correspond to the reported infectives in the publicly available data, and the asymptomatic Infectives (J) which correspond to the unreported infectives. The total infectives are I + J. The rest of the compartments are the symptomatic Recovered (R) and the asymptomatic Recovered (U ). The symptomatic Infectives (I) recover at the rate γ, and the asymptomatic Infectives (J) recover at the rate µ. I recover through isolation in the hospital or at home. On the other hand, the J recover spontaneously. The vaccinated population, (V = κS), is a loss from the susceptible compartment: they are added to the recovered compartments. β(t) is the time-varying transmission rate, it usually depends on the infection vector. In the COVID-19 pandemic, β(t) depends also on contacts between individuals. κ is the average percentage of individuals that are vaccinated daily. ξ represents the probability that an infective individual is reported, while (1 − ξ) is the probability that an infective is an asymptomatic infective. The portion of the total infectives that are symptomatic and reported corresponds to ξ(I + J). On the other hand, (1 − ξ)(I + J) represents the asymptomatic infectives. N represents the total population (2). It is assumed that N does not change throughout the pandemic and that infective individuals are immediately infectious. The dynamics of the interactions between the compartments in Figure 1 can be represented by the following system of ordinary differential equations with time-varying transmission rate β(t).
The continuity equation is given by The initial conditions are denoted by S(t 0 ) = S 0 , where t ≥ t 0 represent time in days and t 0 is the start date of the pandemic in the model. The model parameters are summarized in Table 1. Baseline transmission rate β 0 [0,1) fitted using early data [17,9] Probability that an Infected person is reported ξ [0, 1) constant [16] Proportions of daily vaccinated individuals κ [0, 1) constant [17,4] recovery rate of symptomatic infectives γ [0,1) constant [16] recovery rate of asymptomatic infectives µ [0,1) constant [16]

Time-Varying Transmission Rate
Time-varying transmission rate β(t) in (1) incorporates the impact of public health actions and the public response to the actions [33,2]. The formulation of β(t) in [33] includes temperature. This parameter is not considered in the formulation presented in [2], since there is no evidence that temperature plays a role in the transmission of COVID-19. Early in the transmission of COVID-19, the major public health action was lockdown, which was followed by other measures such as social distancing, contact tracing, masking, early detection of infectives and so on. We chose a formulation of β(t) that strongly reflects the pre and post-lockdown periods. In [11] a sigmoid function is used to model a time-dependent decrease in the transmission of COVID-19. In [16], a piecewise constant function is used to model β(t). A piecewise time-varying transmission rate (3) is used to learn a time-dependent transmission rate β in eq. (1). In [16], the piecewise β(t) is defined as follows, The goal of the parameters q 1 , . . . , q n in (3) is to capture the exponential decrease observed in the transmission rate β(t). We choose M 1 , . . . , M n in order to partition the pandemic timeline, according to the onset of various mitigation measures.
We also formulate β(t) following the approach presented in [17,34]. An exponentially decreasing function is used to represent the transmission rate β(t) in (1) to model the impact of lockdown.
where K signifies the onset of government intervention including isolation, quarantine and lockdown. η is the rate at which human contact decreases. We denote K to be the number of days between the date of the first reported case of COVID-19 and the date lockdown was instituted.
When the transmission rate in (1) is assumed to be constant, (β(t) = β), the basic reproduction number can be given by the ratio of the transmission rate to a weighted sum of the symptomatic and asymptomatic recovery rates. However, we observed that this under-estimate the basic reproduction number (R 0 ) for the asymptomatic-SIR model Equation (1). Assuming a disease-free equilibrium of (1), given by (S * , I * , J * , R * , U * ) = (S 0 , 0, 0, 0, 0) Applying the next generation matrix approach [35], the basic reproduction number (R 0 ) is obtained as the spectral radius of the next generation matrix so that If ξ = 0, R 0 = β/µ, when all the infective population are asymptomatic.
Using data from Italy, South Korea, and the United States starting from the date of the first reported cases in the respective countries to the day before vaccination data were reported. The cumulative infective and recovered population data are observed to be non-exponential whenever a mitigation measure such as a comprehensive lockdown is detected in the data. We take the total population N to be 60.36 × 10 6 , 51.64 × 10 6 , and 328.2 × 10 6 in Italy, South Korea and the USA, respectively. In Figures 3a-5a, M κ is zero and so κ = 0 for all the period from the first reported cases to the day before vaccination data are reported.
In addition to learning the parameters, EINN learns ξ, the probability that an infective is reported. A high value of ξ indicates a large number of reported infectives.
When the transmission rate is time-varying, we use a modified reproduction, which we call the timevarying reproduction R t . This time-varying reproduction number, R t , demonstrates the spread pattern of COVID-19 throughout the duration of the pandemic [16].

Feedforward Neural Network (FNN)
An FNN can be represented as a function of L layers, t input vector and an output N where θ: = (W 1 , . . . , W L , b 1 ,. . . , b L ). W k , k = 1, . . . , L, is the set of the neural network weight matrices while b k , k = 1, . . . , L, is the set of the bias vectors. σ is the activation function. Given a collection of sample pairs (t j , u j ), j = 1, . . . M , where u is some target function. The goal is to find θ * by solving the optimization problem The function 1 M M j=1 ||N (t j ; θ) − u j || 2 2 on the right-hand side of (8) is called the mean squared error (MSE) loss function. A major task in training a network is to determine the suitable number of layers and the number of neurons per layer needed, the choice of activation function, and an appropriate optimizer for the loss function [36].

Epidemiology-Informed Neural Network (EINN)
EINN is a type of Feedforward Neural Network that includes the known epidemiology dynamics in its loss function. In this paper, EINN is adapted for the asymptomatic-SIR model (1), where the Mean Square Error (MSE) of this neural network's loss function includes the known epidemiology dynamics such as a lockdown, while other mitigation measures such as social distancing, and contact tracing are detected by the time-varying transmission rate. The output of EINN are the learned solutions to the asymptomatic-SIR model (1) denoted by S(t j ; θ; λ), I(t j ; θ; λ), J(t j ; θ; λ), R(t j ; θ; λ), U (t j ; θ; λ), j = 1, . . . , M . Where θ represent the neural network weights and biases and λ represent the epidemiology parameters. M is the number of training set. The network representing the time-varying transmission rate is denoted by β(t j ; φ; η), j = 1, . . . , M , The parameter φ represents the weights and biases of this network and η is the exponential decay parameter. The training data are generated using cubic spline and denoted byĨ(t j ), R(t j ), j = 1, . . . , M andṼ (t j ), j = 1, . . . , M κ from the given dataset. Here M κ is the number of vaccination days. We observe that training data are not available for all the compartments in the asymptomatic-SIR model; however, EINN is able to capture the epidemiology interactions between the compartments because the epidemiology model residual is included in the MSE loss function. The MSE loss function for EINN with the time-varying transmission rate is given by where the residual L i , i = 1, . . . 6 is as follows In Figure   The term KP s represent the known dynamics in the transmission rates pattern and ICs represent the initial condition for the asymptomatic population.

Data-Driven Simulation Results for Constant Transmission Rates
Using data from Italy, South Korea, and the United States starting from the date of the first reported cases in the respective countries to the day before vaccination data were reported. The cumulative infective and recovered population data are observed to be non-exponential whenever a mitigation measure such as a comprehensive lockdown is detected in the data. We take the total population N to be 60.36 × 10 6 , 51.64 × 10 6 , and 328.2 × 10 6 in Italy, South Korea and the USA, respectively. In Figures 3a-5a, M κ is zero and so κ = 0 for all the period from the first reported cases to the day before vaccination data are reported.
In addition to learning the parameters, EINN learns ξ, the probability that an infective is reported. High value of ξ indicates large number of reported infectives.      Higher ξ values in Tables 5-7, increase the symptomatic infectives population and reduce the asymptomatic population in general. This is reflected by the increase in the βξ column and the corresponding decrease in the β(1 − ξ) column. This means that more people will be in hospitalization/isolation. This translates to more recovery in the symptomatic compartment. We see that the detection of early infectives alone is not enough to mitigate an infectious disease such as COVID-19 as demonstrated in the R 0 column in Tables 5-7. It should be combined with other measures such as contact tracing of infectives.

Social Distancing
It is widely understood that measures such as a lockdown, social distancing, and widespread adoption of facial coverings result in the mitigation of COVID-19. Social distancing is often the most sought-after    contacts. This is demonstrated by reducing β, the transmission rate [16]. The impact of social distancing on the R 0 is presented in the following Tables 8-10.    Reducing β in Tables 8-10 correspond to a reduced symptomatic infectives population I. There is an increase in the asymptomatic infectives population J. Social distancing is effective when the asymptomatic infective population J diminishes. βξ and β(1 − ξ) both decreases. Social distancing should be combined with contact tracing and early detection of infectives population.

Contact Tracing of Infectives
Contact tracing is equivalent to increasing the symptomatic recovery and asymptomatic recovery rates [16]. However, since we do not have reported data for the asymptomatic population, in this paper, we pursue contact tracing as an increase in the symptomatic recovery rate. This is equivalent to reducing the    The raising of γ in Tables 11-13, increases the symptomatic infectives population I which is demonstrated in increased ξ and increased β. β(1 − ξ) decreases while βξ increases. This also results in a reduced R 0 . Contact tracing is an efficient mitigation measure in lowering the spread of COVID-19.

Data-Driven Simulation Results for Vaccination Efficacy
The mitigation measures described in Section 3.2 are non-pharmaceutical measures. In this Section, we discuss vaccination. In the fight against COVID-19, countries such as USA and United Kingdom began to vaccinate in December 2020. A major goal of vaccination is to reduce the susceptible population, i.e.,  people recover without becoming infected. This constitutes a pharmaceutical mitigation measure. We considered the vaccination data for the USA and the United Kingdom, and simulate the effectiveness of vaccination on the daily reported infectives. A hybrid neural network is used to simulate an efficient vaccination strategy in [38]. We show that the implementation of Algorithm 1 for the asymptomatic-SIR model (1), we can demonstrate the efficacy of vaccination in combination with some mitigation measures.
In Figure 6 we present a simulation of the effectiveness of vaccination in combination with an increase in social distancing in the USA and in the United Kingdom. We used the USA projection of 1, 000, 000 daily vaccination. In the case of the magenta curve, we learned κ using the daily vaccination data. The first reported case was 01/22/2020, Vaccination data were first reported on 19 December 2020. In 6(a) the model is extrapolated for 2 cases. The red curve is the case of no vaccination, here κ = 0. In the magenta curve, we learned κ using the daily vaccination data. The first reported case was on 31 January 2020, Vaccination data were first reported on 13 December 2020. The effectiveness of vaccination is demonstrated by learning the pre-vaccination and post-vaccination epidemiology parameters using smooth daily reported infectious data from the USA. In 6(b) the effectiveness of vaccination is demonstrated by learning the pre-vaccination and post-vaccination epidemiology parameters using smooth daily reported infectives data from the United Kingdom.

Data-Driven Simulation Results for Time-Varying Transmission Rate
In the EINN Algorithm 2, M β corresponds to the number of days mitigation is delayed in the data, which is equal to K in Equation (4). M κ is the number of vaccination days. In Figures 7(a)  in EINN Algorithm 2. We take β 0 = 0.22, obtained using early data and nonlinear regression. EINN Algorithm 2 learns η = 0.87, the rate at which human contact decreases. In 7(b) The delayed-mitigation exponential transmission rate is learned using Equation (4) in Equation (1). We set K = 57 and we fix ξ = 0.46 in EINN Algorithm 2. We take β 0 = 0.279, obtained using early data and nonlinear regression.
EINN Algorithm 2 learns η = 0.60, the rate at which human contact decreases.

Data-Driven Simulation for Piecewise Transmission Rate
In the EINN Algorithm 3, M i , 1 ≤ i ≤ n are chosen to corresponds to a partitioning in the data.
Time-varying transmission rates learned by the EINN Algorithm 3 are presented in Figures 8(a) and 7(b).
For Italy and USA data, we used the following formulation for β(t) in Algorithm 3   4. Discussion

Mitigation Measures
The COVID-19 infectious population surge witnessed in March and April 2020 around the world forced many countries to institute strict lockdown measures. This was largely successful in reducing the R 0 in

Vaccination Efficacy
In Figure 6(b), using USA data, the mitigation effect of vaccination on the daily infectives is demonstrated. Implementing Algorithm 1, we obtained κ = 0.00184, which is slightly different from the projection of κ = 0.00305, corresponding to 1 million people vaccinated per day. In Figure 6(a), using United Kingdom data, we simulate the impact of vaccination on the daily reported infectives, using a smoothed daily vaccination data from 13 December 2020 to 5 February 2020 and smoothed daily reported infectives data.
We implement Algorithm 1 and we obtained κ = 0.00305. We demonstrate the impact of increased social distancing together with the vaccination effort. Social distancing corresponds to decreasing the transmission rate β. Increased social distancing reduces the daily reported infectives but it extends the number of days daily infectives data is significant.

Time-Varying Transmission Rate
In Section 3.4, the delayed-mitigation exponential time-varying transmission rate detects the impact of 2020 COVID-19 lockdown, as well as the other mitigation measures post-lockdown using the parameter η. It is however difficult to know if η captures all the pattern in the time-varying transmission rate as demonstrated in Figure 7a,b, i.e., whether or not Equation (4) helps us to learn the most accurate form of β. For instance, the time-varying basic reproduction rate R t is underestimated pre-lockdown in the USA data and overestimated pre-lockdown in Italy data.

Error Metrics for Data-Driven Simulation
The performance of the neural network training is demonstrated in Table 16, where the random and shuffle splits [39] has been used to generate the training and testing dataset. The random split performed better than the shuffle split. In Figure 9, we present the training and testing MSE at different epochs, depths and widths. We observe that it is more beneficial to increase the width before increasing the depth [40].

Conclusions
We have presented a data-driven deep-learning algorithm that discovers transmission rate patterns in an epidemiology model using cumulative and daily reported symptomatic infective and recovered data. The algorithm predicts asymptomatic infectives and asymptomatic recovered populations. The asymptomatic population is usually unreported in the publicly available data. We learn this population from symptomatic population data. It is demonstrated that a time-varying function models the nonlinear transmission rate.
The EINN algorithms presented, learns the nonlinear time-varying transmission rate without a pre-assumed pattern. This approach is useful when the dynamics of an epidemiological model is impacted by various mitigation measures. The algorithm can be adapted to most epidemiology models. In

Appendix B. EINN Algorithm for Time-Varying Transmission Rate
The time-varying transmission rate is non-constant in the presence of mitigation measures in the cumulative infective data. In [17,4], it was shown that during the early phase of the COVID-19 pandemic when the cumulative infection population grew exponentially, the transmission rate was constant. This coincides with the period before any mitigation measure. Incorporating measures such as social distancing, lockdown and widespread adoption of facial covering in an epidemiology model is complex. We learn an exponentially decreasing transmission rate, we see that it takes the form of Equation (4). Our approach also detects various other post-lockdown mitigation measures. We use EINN Algorithm 2 to learn β(t). Initialize the decay parameters: q 1 , q 2 , q 3 , q 4 , . . . , q n Output layer: β(t j ; φ; q 1 , q 2 , q 3 , q 4 , . . . , q n ) β(t j ; φ; q 1 , q 2 , q 3 , q 4 , . . . , q n ) = β 0 q 2 β(t j ; φ; q 2 ) M 1 < t j ≤ M 2 β 0 q 3 β(t j ; φ; q 3 ) M 2 < t j ≤ M 3 β 0 q 4 β(t j ; φ; q 4 ) M 3 < t j ≤ M 4 . . . β 0 q n β(t j ; φ; q n ) M n < t j ,