Modeling the dynamics of infectious diseases (using COVID-19 as an example)

Modeling the dynamics of infectious diseases (using COVID-19 as an example)

MSC 2020: 92D30, 37N25

Nurgaliev Bakdaulet Erlanuly

East Kazakhstan Technical University named after D. Serikbayev, Ust-Kamenogorsk, Kazakhstan

1Email: bakhascomeback@gmail.com

ANNOTATION This work examines the mathematical modeling of infectious disease spread dynamics using COVID-19 as an example. A modified compartmental SIR model is used, taking into account asymptomatic disease progression and specific mortality rates. The methodology includes time series analysis and the numerical solution of systems of differential equations.

The results demonstrate the impact of various restrictive measures on infection dynamics and the burden on the healthcare system. The scientific novelty lies in adapting the classical SIR model to the characteristics of a specific virus strain. This study allows for estimating the herd immunity threshold and predicting the development of the epidemic, which is important for preparing for future outbreaks of respiratory diseases.

Keywords:mathematical modeling, infectious diseases, COVID-19, SIR, differential equations, forecasting, population dynamics.

     INTRODUCTION

The spread of infectious diseases remains one of the most pressing challenges for modern science and healthcare. The COVID-19 pandemic has demonstrated the need to use mathematical models to analyze and forecast epidemiological processes. Such models make it possible to estimate the rate of infection, predict the development of the epidemic, and analyze the impact of various factors, including human contact and control measures.

Mathematical modeling is an important tool for studying the dynamics of infectious diseases. It allows us to describe infection spread processes using a system of differential equations and predict changes in the size of various population groups over time. Using models, we can study the development of an epidemic at the population level and assess the impact of various parameters, such as infection and recovery rates.

One of the main goals of mathematical modeling is not only forecasting but also understanding the mechanisms of infection spread. However, real epidemiological processes are complex, linked to human behavior, environmental conditions, and disease characteristics. Therefore, models allow for approximate estimates and identification of key patterns in infection spread [1].

The foundations of mathematical modeling of epidemics were laid by William Kermack and Anderson McKendrick, who proposed a compartmental approach that divides the population into groups based on disease status. To analyze an epidemic with a constant population, an approach is used in which the birth and death rates are assumed to be equal, ensuring the stability of the model [2].

This paper examines mathematical modeling of infectious disease dynamics using the SEIR model, using COVID-19 as an example. This model allows for analyzing the spread of infection and forecasting the development of the epidemic.

MATERIALS AND METHODS OF RESEARCH 

The work uses the classical compartmental model of SIR, described by a system of differential equations [3]:

[pic 1]

[pic 2]

[pic 3]

(1)

Where:

S(t) is the number of susceptible individuals

I(t) is the number of infected persons

R(t) is the number of recovered individuals

N is the total population

β is the coefficient of infection transmission

γ is the recovery coefficient

The basic reproductive number is defined as:

[pic 4]

(2)

The first equation represents a decrease in the number of susceptible individuals. S is the number of susceptible individuals, and I is the number of infected individuals. is the transmission rate. The second equation represents a change in the number of infected individuals. There are two processes: an increase ( ) and a decrease ( ), where ( ) is the recovery rate. The third equation represents an increase in the number of recovered individuals.[pic 5]

To perform numerical calculations in Python, the NumPy library is used, which provides a special data type—ndarray (multidimensional array). This data type is designed for efficient storage and processing of numerical information.

To use the library functionality, you need to import the module:

import numpy as np[4].

NumPy arrays are highly computationally efficient and allow mathematical operations to be performed on all elements of the array at once, which is especially important when solving systems of differential equations and modeling epidemiological processes [5]. Machine learning tasks use various basis functions, including polynomial, Gaussian, and sigmoid functions. Polynomial functions have the form

[pic 6]

(3)

but they affect the entire data domain. The sigmoid function is also widely used:

[pic 7]

(4)

which is used in logistic regression to predict the probability of an event. These methods are used to analyze and forecast epidemiological data [6].

In epidemiological modeling, a similar approach is used to describe the interactions between population groups, such as susceptible, infected, and recovered. This allows for a mathematical description of the spread of infection and the prediction of epidemic dynamics [7].

R0 estimates based on the rate of infection growth are more reliable than estimates based on the total number of reported cases, as the latter may be underestimated due to underreporting [8].

[pic 8]

Figure 1.An Algorithm for Implementing the SEIR Model for Predicting the Spread of COVID-19

To visualize the operation of the SEIR model, a flow chart of the algorithm is used (Fig. 1), showing the sequence of calculations and the updating of the states S, E, I and R at each time step.

The aim of this study is to model the spread of an infectious disease, using COVID-19 as an example, using a mathematical model based on a system of differential equations. To achieve this goal, the following tasks were defined: analyzing existing mathematical models of infectious diseases, constructing a SIR-type model, numerically solving the system of differential equations, and analyzing the spread of infection dynamics.