The basic reproductive ratio of Barbour's two-host schistosomiasis model with seasonal fluctuations

Motivated by the first mathematical model for schistosomiasis proposed by Macdonald and Barbour's classical schistosomiasis model tracking the dynamics of infected human population and infected snail hosts in a community, in our previous study, we incorporated seasonal fluctuations into Barbour's model, but ignored the effect of bovine reservoir host in the transmission of schistosomiasis. Inspired by the findings from our previous work, the model was further improved by integrating two definitive hosts (human and bovine) and seasonal fluctuations, so as to understand the transmission dynamics of schistosomiasis japonica and evaluate the ongoing control measures in Liaonan village, Xingzi County, Jiangxi Province.

The basic reproductive ratio R ₀ and its computation formulae were derived by using the operator theory in functional analysis and the monodromy matrix theory. The mathematical methods for global dynamics of periodic systems were used in order to show that R ₀ serves as a threshold value that determines whether there was disease outbreak or not. The parameter fitting and the ratio calculation were performed with surveillance data obtained from the village of Liaonan using numerical simulation. Sensitivity analysis was carried out in order to understand the impact of R ₀ on seasonal fluctuations and snail host control. The modified basic reproductive ratios were compared with known results to illustrate the infection risk.

The Barbour's two-host model with seasonal fluctuations was proposed. The implicit expression of R ₀ for the model was given by the spectral radius of next infection operator. The R ₀ s for the model ranged between 1.030 and 1.097 from 2003 to 2010 in the village of Liaonan, Xingzi County, China, with 1.097 recorded as the maximum value in 2005 but declined dramatically afterwards. In addition, we proved that the disease goes into extinction when R ₀ is less than one and persists when R ₀ is greater than one. Comparisons of the different improved models were also made.

Based on the mechanism and characteristics of schistosomiasis transmission, Barbour's model was improved by considering seasonality. The implicit formula of R ₀ for the model and its calculation were given. Theoretical results showed that R ₀ gave a sharp threshold that determines whether the disease dies out or not. Simulations concluded that: (i) ignoring seasonality would overestimate the transmission risk of schistosomiasis, and (ii) mollusiciding is an effective control measure to curtail schistosomiasis transmission in Xingzi County when the removal rate of infected snails is small.