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Positivity Preserving Truncated Euler-Maruyama Scheme for the Stochastic Age-Structured HIV/AIDS Model
Jie Ren1, Huaimin Yuan2, Qimin Zhang1
1School of Mathematics and Statistics, Ningxia University, Yinchuan, P.R. China.
Summary
Predicting human immunodeficiency virus (HIV) dynamics is challenging. This study develops a numerical method combining Galerkin finite elements and a truncated Euler-Maruyama scheme for accurate HIV model predictions.
Area of Science:
- Mathematical Biology
- Computational Epidemiology
- Numerical Analysis
Background:
- Stochastic age-structured models are crucial for understanding Human Immunodeficiency Virus (HIV) / Acquired Immune Deficiency Syndrome (AIDS) dynamics.
- Analytical solutions for these complex models are often intractable, necessitating robust numerical methods for accurate predictions.
Purpose of the Study:
- To develop and analyze an efficient, fully discrete numerical approximation for stochastic age-structured HIV/AIDS models.
- To assess the accuracy of the proposed numerical scheme by analyzing the error between numerical and analytical solutions.
Main Methods:
- Employed the Galerkin finite element method for discretizing the age variable.
- Utilized a positivity-preserving truncated Euler-Maruyama scheme for time variable discretization.
- Performed theoretical error analysis to validate the numerical scheme's convergence and stability.
Main Results:
- A novel fully discrete numerical scheme was successfully proposed and implemented.
- Theoretical error analysis demonstrated the scheme's capability to approximate the model's behavior.
- Numerical examples confirmed the accuracy and effectiveness of the developed method.
Conclusions:
- The proposed numerical scheme offers an efficient and accurate approach for predicting the dynamic behavior of stochastic age-structured HIV/AIDS models.
- This method overcomes the limitations of analytical solutions, providing a valuable tool for epidemiological research and public health strategies.
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