Dynamics of an influenza epidemic model incorporating immune boosting and Ornstein–Uhlenbeck process

Influenza diseases cause tremendous damage to global public health and economics. The modelling of the spread of influenza has been extensively done, but the population level effect of immune boosting due to repeated infection is poorly understood. In this work, we propose a novel stochastic Susceptible-Infectious-Recovered-partially-protected model to incorporate the effect of immune boosting. At the same time, the transmission rate is modelled by the Ornstein–Uhlenbeck process to capture environmental noise. To govern the threshold dynamics of the stochastic model, a stochastic basic reproduction number R1s is defined. We conclude that if R1s>1, the influenza is almost surely persistent, while if R1s<1, under mild extra conditions, the outbreak of influenza will be suppressed with probability one. Besides, solving the corresponding Fokker–Planck equation gives exact expression of the probability density function for the stationary distribution of the stochastic influenza model. Numerically, we show that large intensity or small reversion favours suppression of influenza disease with probability one, but small intensity or large reversion favours the persistence thus mitigation measure should be strengthened. These findings have important practical implications.