Forecasting the trend of COVID-19 under Imperfect Vaccination using the Homotopy Perturbation Method

Since its onset in late 2019, the COVID-19 pandemic, which was brought on by the new SARS-CoV-2 virus, has seriously upended economies, societies, and healthcare systems throughout the world. In order to control and comprehend the virus's quick spread, A new mathematical model is created, analysed and used to forecast the time growth of COVID-19 under the consequence of imperfect vaccination. The qualitative analysis involves establishing the model's positive invariant region and its solution. The model was confirmed to be linearly stable at the steady states of disease-free and endemic. Analysis of sensitivity, based on basic reproduction number, revealed that transmission, vaccination, and recruitment rates are key factors in COVID-19 spread, with vaccination reducing transmission by 74% and the transmission rate lowering it by 95%. Using the homotopy perturbation method, the model was semi-analytically solved, and results simulated with Maple software indicated a high probability of reducing the susceptible population to zero within a year of widespread vaccine coverage. However, waning immunity due to imperfect vaccination could reverse this trend within a year and a half, increasing susceptibility and decreasing recovery rates. Therefore, booster vaccines are crucial for sustaining immunity, controlling exposure, and managing long-term disease transmission.