In a long-lasting major disease outbreak such as that of COVID-19, the challenge for public health authorities is to keep people motivated and keen on following safety guidelines. In this study, a compartmental model with a heterogeneous transmission rate (based on awareness) is utilized to hypothesize about the public adoption of preventive guidelines. Three subsequent outbreaks in South Korea, Pakistan, and Japan were analyzed as case studies. The transmission, behavior change, and behavioral change ease rates of the disease were measured in these countries. The parameters were estimated using the maximum likelihood method with an additional identifiability analysis performed to determine the uniqueness of the estimated parameters for quantitatively comparing them during the first three waves of COVID-19. The mathematical analysis and simulation results show that individual responses had a significant effect on the outbreak. Individuals declining to follow the public health guidelines in Korea and Japan between the second and third waves contributed to making the third peak the highest of the three peaks. In Pakistan, however, individual responses to following public health guidelines were maintained between the second and third waves, resulting in the third peak being lower than the first, rather than being associated with the highest transmission rate. Thus, maintaining a high level of awareness is critical for containing the spread. Improvised public health campaigns are recommended to sustain individual attention and maintain a high level of awareness.

In this article, we focus on association schemes with some properties derived from the orbitals of a transitive permutation group G with a one-point stabilizer H satisfying H < N G (H) < N G (N G (H)) G and |N G (N G (H))| = p 3 where p is a prime. By a corollary of our main result we obtain some inequality which corresponds to the fact |G : N G (N G (H))| ≤ p + 1.