Exploring the dynamics of monkeypox transmission with data-driven methods and a deterministic model

Abstract

Introduction: Mpox (formerly monkeypox) is an infectious disease that spreads mostly through direct contact with infected animals or people's blood, bodily fluids, or cutaneous or mucosal lesions. In light of the global outbreak that occurred in 2022-2023, in this paper, we analyzed global Mpox univariate time series data and provided a comprehensive analysis of disease outbreaks across the world, including the USA with Brazil and three continents: North America, South America, and Europe. The novelty of this study is that it delved into the Mpox time series data by implementing the data-driven methods and a mathematical model concurrently-an aspect not typically addressed in the existing literature. The study is also important because implementing these models concurrently improved our predictions' reliability for infectious diseases.

Methods: We proposed a traditional compartmental model and also implemented deep learning models (1D- convolutional neural network (CNN), long-short term memory (LSTM), bidirectional LSTM (BiLSTM), hybrid CNN-LSTM, and CNN-BiLSTM) as well as statistical time series models: autoregressive integrated moving average (ARIMA) and exponential smoothing on the Mpox data. We also employed the least squares method fitting to estimate the essential epidemiological parameters in the proposed deterministic model.

Results: The primary finding of the deterministic model is that vaccination rates can flatten the curve of infected dynamics and influence the basic reproduction number. Through the numerical simulations, we determined that increased vaccination among the susceptible human population is crucial to control disease transmission. Moreover, in case of an outbreak, our model showed the potential for epidemic control by adjusting the key epidemiological parameters, namely the baseline contact rate and the proportion of contacts within the human population. Next, we analyzed data-driven models that contribute to a comprehensive understanding of disease dynamics in different locations. Additionally, we trained models to provide short-term (eight-week) predictions across various geographical locations, and all eight models produced reliable results.

Conclusion: This study utilized a comprehensive framework to investigate univariate time series data to understand the dynamics of Mpox transmission. The prediction showed that Mpox is in its die-out situation as of July 29, 2023. Moreover, the deterministic model showed the importance of the Mpox vaccination in mitigating the Mpox transmission and highlighted the significance of effectively adjusting key epidemiological parameters during outbreaks, particularly the contact rate in high-risk groups.

Keywords: Mpox; deterministic model; epidemiology; neural networks; univariate time series.

Figure 1

Distribution of the global population sizes, Mpox cases, and incidence: Panel (A) displays the population of each continent, where the world population is 7,795,237,030, and panel (B) shows the distribution of total Mpox cases in seven continents by the pie chart. Panel (C) presents incidence per 100 k individuals across the seven continents, where the maximum incidence occurs in North America (6.26 per 100 k), South America (5.19 per 100 k), and Europe (3.43 per 100 k). Panel (D) shows that Spain has a higher incidence of Mpox cases (greater than 15 per 100 k) compared to the other countries. Specifically, USA, Portugal, and Luxembourg maintain an incidence rate of 8 cases per 100 k, while the incidence rate in the remaining countries is below 8 per 100 k.

Figure 2

Preparing time series data: This figure shows five important phases: data pre-processing steps (panels A and B, top left), weekly time series of reported cases (panel C, top right), weekly time series of reported cases on a normalized scale (panel D bottom right), and the decomposition of the confirmed cases to check seasonality and stationarity (panel E, bottom left). Given the ADF Statistic of $-$2.04679, which is negative, and a $p$-value of 0.26644, exceeding the significance threshold of 0.05, we can infer that the EPI weekly worldwide data exhibits non-stationarity. Panel E also demonstrates the seasonality patterns, as the seasonal decomposition shows the rise and fall, with a period of seven. We observe an increasing trend in the number of reported cases until the EPI week of 14 (between July and August, during the summer), then started decreasing. Panel E displays the additive decomposition, where the seasonal chart scales between $-$100 to 100 and the trend chart scales between $-$1,000 to 1,000.

Figure 3

Visualization of globally reported Mpox cases: Panel (A) displays the confirmed Mpox cases across 112 countries using a treemap visualization. Panel (B) shows the global and regional spread of confirmed Mpox cases on a world map. Among all countries, the USA has recorded the highest number of cases (30,225), followed by Brazil (10,941), Spain (7,555), France (4,146), Colombia (4,090), and Mexico (4,017).

Figure 4

Visualization the spread of Mpox cases in the USA: Panel (A) displays the distribution of Mpox cases (left top) across the US map (35) . Panel (B) shows the time series of confirmed Mpox cases (right top) from week 0 to 51, covering the period from June 4, 2022, to May 27, 2023. Panel (C) illustrates the states with the highest incidence of Mpox per 100,000 (100 k) people in the United States. This panel also shows that the District of Columbia (77.01% per 100 k) has higher Mpox incidence from May 2022 to May 2023. Next, the other states such as New York (18.61% per 100 k), California (14.58% per 100 k), Florida (13.44% per 100 k), Maryland (12.1% per 100 k), Illinois (11.48% per 100 k), Nevada (10.04% per 100 k), and Texas (10.33% per 100 k) are the higher risk of an epidemic. Additionally, Washington (9.06% per 100 k), Arizona (8.28% per 100 k), and New Jersey (25.57% per 100 k) have a higher chance of an epidemic compared to the other states.

Figure 5

Data partitioning into training, validation, and testing sets: This figure illustrates the decomposition of the training, validation, and testing data based on the EPI week for deep learning (1D-CNN, LSTM, BiLSTM, hybrid CNN-LSTM, and CNN-BiLSTM), statistical (ARIMA and exponential smoothing), and deterministic models. Note that data from May 01 to May 31, 2023, was accessed on June 03, 2023. Later, new data for comparing model predictions were accessed on August 01, 2023.

Figure 6

Building Blocks of deterministic and data-driven models: (A) displays different methods and demonstrates how to build the predictive models. (B) identifies the best predictive models by evaluating those models.

Figure 7

Building Blocks of data-driven models: This figure was adapted from Crick (41), illustrating how a model optimizer is generally used in model training. The loss function $L(y,\acute{y})$ is computed by comparing predicted values $(\acute{y})$ and true values $(y)$, and model parameters are adjusted to minimize $L(y,\acute{y})$ that enhances the accuracy of prediction model $f$.

Figure 8

Mpox model sensitivities to its associated parameters for six different locations: Panel (A) and (B) illustrate the sensitivity index of ${\mathcal{R}}_0$ (Equation 13) for $\tau =0$ and $\tau =0.01$, respectively. These are computed with the fitted parameters described in Table 3, variable parameters listed in Table 4, and remaining parameters given in Table 2. The figure demonstrates that the sensitivity index of the parameters ${\beta }_{hh},{\theta }_h,{\gamma }_1,{\mu }_h,\tau$, and $\varphi$ changes when the parameter values change, while the remaining parameters remain unchanged. In other words, the parameters ${\alpha }_h,{\gamma }_h,{\delta }_h,\sigma$, and $h$ are almost constant and do not depend on another parameter, but the remaining parameters show the dependence on a second parameter, affecting the varying sensitivity index. The simulation depicts the sensitivity index of ${\mathcal{R}}_0$ (Equation 13) for six different locations using six different colors and sizes of circles.

Figure 9

Neural network architecture: CNN, LSTM, BiLSTM, CNN-LSTM, and CNN-BiLSTM architectures involve neural network layers for feature extraction on input data. Model descriptions for time series prediction problems from sequences of data are provided in the Appendix (Tables A4–A9).

Figure 10

Data fitting for the Mpox deterministic model in various locations: The presented figure illustrates the Mpox data fitting of the deterministic model in various locations. As a mathematical model of the Mpox spread, we use the deterministic model shown in Figure A2. Also, Table 2 provides the initial data used for the fitting, while Table 3 contains the corresponding fitting parameters for each location. Additionally, Table 4 shows various variable parameters using the testing errors defined in Equation 2.

Figure 11

Importance of Vaccinations: This figure illustrates the significant impact of vaccination on the USA population size $N_h=331\text{,}002\text{,}651$. Panel (A) depicts the percentage of epidemic sizes for ${\varphi }_0\in [0,0.25]$. Panel (B) demonstrates a decreasing fraction of incidence with increasing vaccination rates. Panels (C) and (D) depict the cumulative incidence and the prevalence, respectively, highlighting a substantial reduction when the vaccination rate ${\varphi }_0$ increases. Overall, raising the vaccination rate $\varphi$ of Mpox offers a solution to mitigate the risks posed by Mpox as well as other emerging infectious diseases. For these simulations, we use the parameter values given in Table 2 except the variable parameter values ${\beta }_{hh}=5.5,{\beta }_{rr}=2.5,{\beta }_{hr}={\beta }_{rh}=2.0251$. For each simulation, we also consider the initial conditions $I_h(0)=I_r(0)=1,S_h(0)=N_h-I(0)-V_1(0),E_h(0)=H(0)=V_1(0)=V_2(0)=R_h(0)=E_r(0)=R_r=0,S_r(0)=N_r-I_r(0)$, where $N_r=8\text{,}000\text{,}000$. We run simulations over the time range $\tau =[0,65]$ with step size $\mathrm{\Delta }\tau =0.1$.

Figure 12

Epidemic (yellow colored) vs. non-epidemic (blue colored) parametric regions depending on $c_0$ and $k_1$: The figure shows the colormaps for the range of $c_0=[0,1]$, and $k_1=[0,1]$ with step size $\mathrm{\Delta }{c}_0=\mathrm{\Delta }{k}_1=0.1$. The epidemic region is depicted in yellow in the figure, while the non-epidemic parametric region is represented in blue. For these simulations, we use the parameter values given in Table 2, fitted parameter in Table 3, and variable parameters value in Table 4. This figure also illustrates that with these epidemiological parameters, it is possible to control the epidemic by adjusting the baseline contact rate $c_0$ and the proportion of contacts within the human population $k_1$ in overall contacts. For each simulation, we also consider the initial conditions $I_h(0)=I_r(0)=1,V_1(0)=N_h\ast (1/100),S_h(0)=N_h-I(0)-V_1(0),E_h(0)=H(0)=V_2(0)=R_h(0)=E_r(0)=R_r=0,S_r(0)=N_r-I_r(0)$, where $N_r=8\text{,}000\text{,}000$. We run simulations over the time domain $\tau =[0,65]$ with step size $\mathrm{\Delta }\tau =0.01$.

Figure 13

Deep learning model training for Mpox incidence in different locations: This figure shows the predicted vs. actual incidence per week in six different locations worldwide using five different deep learning models. We use 60% for the training dataset with 10% for validation during the model building (details given in the appendix) and then utilize the trained model to predict the incidence (number of new cases) per week for the entire time series data spanning from 2023-02-11 to 2023-06-03. For example, in the World and Europe time series data, the model was trained on the data from 2022-05-07 to 2023-02-04 (week 0 to week 39), and the predictions were then extended to cover the entire time whole period from 2022-05-07 to 2023-06-03 (week 0 to week 56) and compared with the actual data. The actual data are displayed in red in the panel, while the predictions from various deep learning models, including CNN, LSTM, BiLSTM, CNN-LSTM, and CNN-BiLSTM, are presented in magenta, green, blue, black, and cyan, respectively. These models were also evaluated using test data from 2023-02-11 to 2023-06-03 (week 40 to week 56), with the model predictions given in Figure 14 and subsequently used to predict the next eight weeks from 2023-06-10 to 2023-07-29 (week 57 to week 64) in Figure 15. Similarly, we optimized the model for other locations as provided in the Appendix C.

Figure 14

Data driven model on test data for Mpox incidence across different locations.

Figure 15

Mpox predictions of different locations using data-driven and deterministic models. Appendix C contains the data-driven model construction process, along with error measurements on the test data. Additional details, i.e., model parameters and model construction are provided in Tables A1, A2, A3, A4, A5, A6, A7, A8, and A9. Moreover, the deterministic model used the same parameter values described in Figure 10.

Figure A1

Dynamics of daily Mpox incidence: This figure illustrates the daily incidence of Mpox in various locations from May 2022 to May 2023. The figure exhibits seasonality and includes missing reported data. In contrast, the weekly data in Figure 2C captures the proper disease dynamics, which can be a pragmatic choice.

Figure A2

Schematic representation of the deterministic model: This figure shows a modified SEIR model with two-phase vaccinations (smallpox or dose one vaccine and dose two vaccines) for mpox virus transmissions.