We conditionally extend formulas of(Dietz and Schenzle, J. Math. Biol. 22: 117-120,1995) for the ”transmission potential” of an immunizing infection with pre-infection possible before birth and vertical transmission admitted. We look for minimum proportion to be covered to reduce basic reproduction rate below 1 by acting through vaccination. We present also a new criterion allowing the selection of an immunizing vaccination strategy by bringing the reproduction number below 1. We find that reduce vertical transmission, adds chances to eradicate disease. Moreover reduce age of vaccination reduces the minimum vaccination coverage inducing global immunization against disease by bringing down the basic reproduction number.

This study proposes and analyzes a mathematical model for the transmission dynamics of COVID-19, explicitly accounting for the ability of the immune system in some individuals to eliminate the virus before they become infectious. A compartmental Susceptible–Exposed–Asymptomatic–Symptomatic–Hospitalized–Recovered (SEAIHR) model is formulated using both classical integer-order derivatives and Caputo fractional-order derivatives. The model is first examined by establishing the positivity and boundedness of solutions, followed by the computation of the basic reproduction number R0. The existence of equilibrium points is proven, and the asymptotic stability of the disease-free equilibrium is analyzed when R0 < 1. Epidemiological data from Cameroon are used to estimate model parameters and calibrate the system. Sensitivity analysis identifies the most influential parameters governing disease transmission. Numerical results indicate that R0 ≈ 1.6483, suggesting that COVID-19 remains endemic despite declining mortality. Simulations further show that enhanced immune responses may contribute to the relatively low morbidity observed in sub-Saharan Africa.

Powdery Mildew is a global plant disease caused by fungal species, causing powdery growth on various parts of plants. This study aims to develop, evaluate and simulate the transmission dynamics of Powdery Mildew in cashew plants using a stochastic differential equation with Lévy noise. After providing some preliminary definitions of stochastic differential equations, we first consider the model without noise. We prove positivity, compute the basic reproduction number, ${\mathcal{R}}_0$, the PMD-free equilibrium, and the existence of a unique endemic equilibrium point whenever ${\mathcal{R}}_0>1$. After that, we formulate the stochastic model under Lévy noise. For this model, we also prove the positivity of the solutions and show that it is possible to extend the disease when ${\mathbb{S}}_s<1$. We also found the condition that ensures the persistence of the disease if ${\mathbb{S}}_0^s>1$. To simulate the model, we build a stochastic model numerical scheme and do a number of numerical simulations to support the theoretical findings we have gotten.