The increasing complexity of IoT networks heightens the risk of malware propagation, requiring efficient mitigation strategies. Traditional cybersecurity solutions are often reactive and do not optimize resource allocation for malware containment. This study formulates an optimal control problem for malware spread using a Susceptible-Infected-Recovered (SIR) model, combining two strategies: reducing transmission and accelerating removal. The control problem is modeled with the Hamilton-Jacobi-Bellman (HJB) equation and solved numerically. The results show that optimal preventive and remedial actions vary with network conditions: high infection levels call for stronger prevention, while widespread infections require intensive remediation. Compared to conventional approaches, this method improves cost efficiency and resource allocation, demonstrating the scalability and effectiveness of HJB-based strategies for IoT cybersecurity.