Dynamic strategies and optimal control analysis for hepatitis C management: non-invasive liver fibrosis diagnosis

This study proposes a novel model employing nonlinear ordinary differential equations to dissect HCV dynamics. Six distinct population groups are delineated: Susceptible, Treatment, Responder, Non-Responder, Cured, and Fibrosis. A detailed numerical analysis of this model was conducted, tracking the predicted trends over a span of 20 years. The primary objective of this analysis is to assess and confirm the model's predictive accuracy and its potential to supplant invasive diagnostic methods in monitoring the progression of liver fibrosis. By incorporating various control parameters, namely <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msub></mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math> the model offers a nuanced perspective on disease progression and treatment outcomes. Parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> modulates treatment-induced fibrosis progression, providing a crucial lever for mitigating treatment-related side effects. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> reflects treatment effectiveness, capturing the proportion of responders within the treatment cohort. Meanwhile, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msub></mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> governs fibrosis progression in non-responders, shedding light on the disease's natural trajectory without effective treatment.