Dynamics of a periodic tick-borne disease model with co-feeding and multiple patches

Abstract

By extending a mechanistic model for the tick-borne pathogen systemic transmission with the consideration of seasonal climate impacts, host movement as well as the co-feeding transmission route, this paper proposes a novel modeling framework for describing the spatial dynamics of tick-borne diseases. The net reproduction number for tick growth and basic reproduction number for disease transmission are derived, which predict the global dynamics of tick population growth and disease transmission. Numerical simulations not only verify the analytical results, but also characterize the contribution of co-feeding transmission route on disease prevalence in a habitat and the effect of host movement on the spatial spreading of the pathogen.

Appendix

We show parameter values used in Fig. 5, including recruitment rates of larval ticks $\rho _k$, probability of co-feeding transmission $\eta _k$, systemic transmission probability between infected rodents and susceptible larvae $\zeta _k^L$ and between susceptible rodents and infected nymphs $\zeta _k^H$, and initial rodent densities $H_k(0)$ in Table 3. Host migration proportions from patch j to patch k take the form $m_{kj}(t)=(m_{kj}^{(1)}~m_{kj}^{(2)})(1 ~ \cos (2\pi t/365))^T$, $j,k=1,2,\ldots ,9$. Table 4 lists all the components $m_{kj}^{(1)}$ and $m_{kj}^{(2)}$ of host migration proportions. Other parameters among the 9 patches are fixed at the same values as follows:

Fig. 5

The comparison of accumulated infected nymph density for model (3) in a year: a 9 patches are isolated from each other; b host population can move freely among 9 patches

Table 3 Different parameter values of model (1) with 9 patches

Table 4 Parameter values of host migration proportion $(m_{kj}^{(1)} \quad m_{kj}^{(2)})^T$

Table 5 Net reproduction numbers and basic reproduction numbers for 9 patches

Table 6 Accumulated yearly nymphal ticks (AYNT) and accumulated yearly infected nymphal ticks (AYINT) with and without migration, and their comparisons for 9 patches $(\times 10^5)$

$$\begin{aligned}&d_k^L=0.01, \quad d_k^N=\left\{ \begin{array}{ll} 0.03-0.01\sin (2\pi t/365), \quad k=1,2,3,4,\\ 0.02-0.01\sin (2\pi t/365),\quad k=5,6,7,8,9,\end{array}\right. \\&d_k^A=0.00625, \quad d_k^H=0.01, \quad \mu _k^L=\mu _k^N=\mu _k^A=0.00001, \\&\zeta _k^L=0.5, \quad \zeta _k^N=0.5, \quad \zeta _k^H=0.57, \\&m_k^L=0.35, \quad m_k^N=0.1, \quad D_k=20. \end{aligned}$$

Without migration, we calculate the net reproduction numbers, the basic reproduction numbers and accumulated infected nymphal densities in 1 year for each patch k, $k=1,2,\ldots , 9$ and list them in Table 5. When host migration is considered, the added numbers of accumulated yearly nymphal ticks and infected nymphal ticks across 9 patches are summarized in Table 6.