# A mathematical study for psoriasis transmission with immune-mediated time delays and optimal control strategies

## TL;DR

This paper uses a mathematical model to study how immune delays affect psoriasis and how biologic treatments can help manage the disease.

## Contribution

The study introduces a novel mathematical model with immune-mediated time delays and optimal control strategies for psoriasis.

## Key findings

- Immune-mediated delays can cause instability and oscillations in psoriasis dynamics through Hopf bifurcations.
- Optimal control strategies using TNF-α and IL-17 inhibitors can stabilize the immune response and reduce keratinocyte density.
- Including time delays in the model provides a more realistic understanding of psoriasis progression and treatment.

## Abstract

In psoriasis, dendritic cells activate T cells, which then release excessive pro-inflammatory cytokines, leading to abnormal growth of keratinocytes in the epidermis. At the same time, anti-inflammatory cytokines attempt to restore balance. In reality, these immune processes are not immediate; they involve biological time gaps due to signal processing, cell communication, and cytokine feedback. Such immune-related delays may play a key role in triggering unstable or oscillatory behavior observed in psoriasis flare-ups. In this study, we present and analyze a mathematical model of psoriasis that explicitly includes two intracellular immune-mediated time delays to demonstrate their biological significance in disease progression. The model captures the interactions among T cells, dendritic cells, keratinocytes, and local mature stem cells. It features two cytokine-mediated feedback loops between T cells and dendritic cells, while stem cells attempt to regulate the immune response through anti-inflammatory signaling. A key challenge is identifying the critical time delays that modulate these interactions. To address this, we introduce two different delays in different interaction terms of the model system. We test the hypothesis that these delays can critically influence the onset and persistence of psoriatic pathology mathematically. Using stability analysis of the interior equilibrium, we determine parametric relations, their ranges, and delay thresholds that give rise to Hopf bifurcations, thereby linking delays to disease and deriving conditions of instability. Our analysis demonstrates that both immune-mediated delays critically influence system stability, with threshold values of τ1* and τ2* inducing oscillations through Hopf bifurcations. Further, we apply optimal control strategies on the delayed system using the effects of two biologic agents: TNF-α and IL-17 inhibitors. Incorporation of optimal controls effectively stabilizes the immune response. Numerical simulations support these analytical findings and show that biologic interventions can effectively reduce keratinocyte density. Inclusion of immune-related delays, based on both analytical and numerical results, provides a more realistic understanding of psoriasis dynamics and helps optimize therapeutic approaches for psoriasis management.