Deterministic and stochastic effects in spreading dynamics: A case study of bovine viral diarrhea
Galler, Markus; Lüdge, Kathy; Humphries, Rory; Mulchrone, Kieran; Hövel, Philipp
Bovine viral diarrhea (BVD) is a disease in cattle with complex transmission dynamics that causes substantial economic losses and affects animal welfare. The infection can be transient or persistent. The mostly asymptomatic persistently infected hosts are the main source for transmission of the virus. This characteristic makes it difficult to control the spreading of BVD. We develop a deterministic compartmental model for the spreading dynamics of BVD within a herd and derive the basic reproduction number. This epidemiological quantity indicates that identification and removal of persistently infected animals is a successful control strategy if the transmission rate of transiently infected animals is small. Removing persistently infected animals from the herd at birth results in recurrent outbreaks with decreasing peak prevalence. We propose a stochastic version of the compartmental model that includes stochasticity in the transmission parameters. This stochasticity leads to sustained oscillations in cases where the deterministic model predicts oscillations with decreasing amplitude. The results provide useful information for the design of control strategies.
Dynamical systems are often described by deterministic mathematical models, where the state of the system is determined by the initial conditions. Many real-world processes, however, include an element of probability or randomness. Even so, deterministic models might still be able to reproduce the main trend as a mean-field approximation but fail to capture the spectrum of possible dynamical scenarios of individual realizations. In addition, stochastic input such as noise can trigger the emergence of hidden dynamical features with surprising effects such as stochastic resonance, coherence resonance, or other noise-induced changes of dynamical behavior. Here, we present the example of a cattle disease that is realized as an extended susceptible-infected-recovered model. To explore the impact of stochasticity on the temporal behavior of the dynamics, we consider a stochastic transmission coefficient and systematically investigate the interplay between parameter noise and the intrinsic time scales of the underlying deterministic system.
