1. School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710119, People's Republic of China
2. School of Mathematics, Northwest University, Xi'an 710127, People's Republic of China
3. Natural Resources Institute, University of Greenwich at Medway, Central Avenue, Chatham Maritime, Kent ME4 4TB, UK
4. Department of Infectious Disease Epidemiology, School of Public Health, Imperial College London, St Mary's Campus, Norfolk Place, London W2 1PG, UK