This is the report from a BSPP Undergraduate ‘Vacation’ Bursary.
Click here to read more/apply for one yourself.
This summer, I had the privilege of working in the Theoretical and Computational Epidemiology lab within the Department of Plant Sciences at the University of Cambridge, under the supervision of Professor Nik Cunniffe.
Traditionally, epidemiological models have employed a simplified approach in which an individual’s infection status is represented by different disjoint stages (e.g., infected or not). Events such as disease transmission or death occur at a constant rate, implying that the time spent in each stage is exponentially distributed. This type of model is usually depicted using arrows connecting boxes representing infection stages. However, assuming constant rates of movement between stages may not reflect reality accurately. In many plant diseases, for example, the latent period follows a gamma distribution. Additionally, for models which track individual plants or trees as the host unit, infectivity or disease severity will increase as a function of time since infection. During my internship, I constructed epidemiological models of citrus canker, a disease caused by the bacterium Xanthomonas citri subsp. citri. By systematically accounting for infectivity and detectability which change over time, the idea was to assess the validity of simplified models in which these features were not present.
To account for time-dependent infectivity, I initially proposed a disease model with logistic growth in infectivity. Recognising that directly modelling this change would be computationally intensive, I approximated the disease progression using multiple compartments, each with distinct infectivity levels. This approach allowed me to approximate the expected infectivity as a logistic curve and compare it with results obtained using the traditional SEI (Susceptible, Exposed, Infectious) model, with the inverse of the transition rate between Exposed and Infectious stages representing the latent period.
Additionally, to assess the impact of detectability, I compared a model that accounted for changing detectability with the SCI (Susceptible, Cryptic, Infectious) model. I then compared the optimal control strategies yielded from the two models.
In summary, I discovered that while the SEI model is generally effective in capturing complex biological processes and making accurate predictions, some conceptual adjustments may be necessary. Specifically, when infectivity increases over time, the parameter corresponding to the “latent period” should be set larger than the actual latent period for predictions to align with the more “realistic” model. The length of time spent in this compartment should thus be regarded as a mere parameter rather than a biologically significant quantity. Furthermore, I found that introducing time-dependent infectivity does not alter the optimal control strategy, but it does influence the expected outcomes.
Throughout this summer internship, I gained valuable experience in translating biological problems into mathematical equations and converting them into computer programmes. This experience has deepened my passion for using mathematics to address biological challenges. I extend my heartfelt gratitude to BSPP for providing this incredible opportunity; without their support, none of this would have been possible.
Wayne Liang
The University of Cambridge
Models for changing infectivity. In both case the models are fitted to a “real” logistic curve. a) In the SEI model, due to its simplicity, parameters are fitted directly to minimise the squared difference between the real infectivity curve and the approximation generated by the model. For the more realistic model, the logistic increase in infectivity is approximated by multiple stage of infection. b) A step function is first fitted to best approximate the logistic curve. By using the length of each step as the mean time spent in each stage of infection, c) the probability of being at each compartment at given time after infection is generated. d) This is then used to fit infectivity of each stage to minimise the squared difference between the approximate and the logistic curve.
A simulated epidemic. Susceptible individuals are coloured in green, Cryptic in blue and Infected in red.
This is the report from a BSPP Undergraduate ‘Vacation’ Bursary.
Click here to read more/apply for one yourself.
