A study of qualitative correlations between crucial bio-markers and the optimal drug regimen of Type I lepra reaction: A deterministic approach

Abstract

Mycobacterium leprae is a bacterium that causes the disease leprosy (Hansen’s disease), which is a neglected tropical disease. More than 2,00,000 cases are being reported per year worldwide. This disease leads to a chronic stage known as lepra reaction that majorly causes nerve damage of the peripheral nervous system leading to loss of organs. The early detection of this lepra reaction through the level of bio-markers can prevent this reaction occurring and the further disabilities. Motivated by this, we frame a mathematical model considering the pathogenesis of leprosy and the chemical pathways involved in lepra reactions. The model incorporates the dynamics of the susceptible Schwann cells, infected Schwann cells, and the bacterial load and the concentration levels of the bio-markers interferon- γ \hspace{0.1em}\text{interferon-}\hspace{0.1em}\gamma , tumor necrosis factor- α \hspace{0.1em}\text{tumor necrosis factor-}\hspace{0.1em}\alpha , IL (interleukin)- 10 \hspace{0.1em}\text{IL (interleukin)-}\hspace{0.1em}10 , IL- 12 \hspace{0.1em}\text{IL-}\hspace{0.1em}12 , IL- 15 \hspace{0.1em}\text{IL-}\hspace{0.1em}15 , and IL- 17 \hspace{0.1em}\text{IL-}\hspace{0.1em}17 . We consider a nine-compartment optimal control problem considering the drugs used in multi drug therapy (MDT) as controls. We validate the model using 2D heat plots. We study the correlation between the bio-markers levels and drugs in MDT and propose an optimal drug regimen through these optimal control studies. We use the Newton’s gradient method for the optimal control studies.