Richards model revisited: Validation by and application to infection dynamics

• Published in: Journal of theoretical biology 2012-11, Vol.313, p.12-19
• Main Authors: Wang, Xiang-Sheng, Wu, Jianhong, Yang, Yong
• Format: Article
• Language: English
• Subjects: Asymptotic approximation; Basic reproduction number; Biological Sciences; biologists; Canada - epidemiology; Communicable Diseases - epidemiology; dengue; Dengue - epidemiology; Disease Susceptibility; disease transmission; Humans; Influenza A Virus, H1N1 Subtype; Influenza, Human - epidemiology; Influenza, Human - virology; Models, Biological; Numerical Analysis, Computer-Assisted; Over fitting; Pandemics; Reproducibility of Results; Richards model; SARS coronavirus; Severe Acute Respiratory Syndrome - epidemiology; Severe Acute Respiratory Syndrome - virology; Singapore - epidemiology; SIR model; Taiwan - epidemiology
• ISSN: 0022-5193; 1095-8541
• DOI: 10.1016/j.jtbi.2012.07.024

Ever since Richards proposed his flexible growth function more than half a century ago, it has been a mystery that this empirical function has made many incredible coincidences with real ecological or epidemic data even though one of its parameters (i.e., the exponential term) does not seem to have clear biological meaning. It is therefore a natural challenge to mathematical biologists to provide an explanation of the interesting coincidences and a biological interpretation of the parameter. Here we start from a simple epidemic SIR model to revisit Richards model via an intrinsic relation between both models. Especially, we prove that the exponential term in the Richards model has a one-to-one nonlinear correspondence to the basic reproduction number of the SIR model. This one-to-one relation provides us an explicit formula in calculating the basic reproduction number. Another biological significance of our study is the observation that the peak time is approximately just a serial interval after the turning point. Moreover, we provide an explicit relation between final outbreak size, basic reproduction number and the peak epidemic size which means that we can predict the final outbreak size shortly after the peak time. Finally, we introduce a constraint in Richards model to address over fitting problem observed in the existing studies and then apply our method with constraint to conduct some validation analysis using the data of recent outbreaks of prototype infectious diseases such as Canada 2009 H1N1 outbreak, GTA 2003 SARS outbreak, Singapore 2005 dengue outbreak, and Taiwan 2003 SARS outbreak. Our new formula gives much more stable and precise estimate of model parameters and key epidemic characteristics such as the final outbreak size, the basic reproduction number, and the turning point, compared with earlier simulations without constraints. ► Revisit Richards model from SIR model. ► Four parameters in either Richards or SIR model are redundant in data fitting. ► Propose a constraint to tackle the over fitting problem. ► Conduct numerical simulations for real time prediction of several diseases. ► Provide stable forecast of final outbreak size and basic reproduction number.