Invasion Speed in Cellular Automaton Models for T. cruzi Vector Migration

Abstract

The parasite Trypanosoma cruzi, known for causing Chagas’ disease, is spread via insect vectors from the triatomine family. T. cruzi is maintained in sylvatic vector-host transmission cycles in certain parts of the Americas. Communication between the cycles occurs mainly through movement (migration) of the insect vectors. In this study, we develop a cellular automaton (CA) model in order to study invasion of a hypothetical strain of T. cruzi through the region defined by the primary sylvatic cycles in northern Mexico and parts of the southeastern United States. The model given is a deterministic CA, which can be described as a large metapopulation model in the format of a dynamical system with 9,376 equations. The migration rates in the model, used as coupling parameters between cells in the CA, are estimated by summing up the proportion of vectors crossing patch boundaries (i.e., crossing from one cell to another). Specifically, we develop methods for estimating speed and direction of invasion as a function of vector migration rates, including preference for a particular direction of migration. We develop two methods for estimating invasion speed: via orthogonal local velocity components and by direct computation of magnitude and direction of an overall velocity vector given a front created by cells identified as being invaded by the epidemic. Results indicate that invasion speed is greatly affected by both the physical and the epidemiological landscapes through which the infection wave passes. A power-law fit suggests that invasion speed increases at slightly less than the square root of increases in migration rate.

Appendix: Equations and Assumptions

The system contains 9,376 equations. Each cell contains anywhere from 4 to 8 equations, depending on the specific patch location. The equations in (1) are representative of a patch 2 interior cell, which contains both species of vector and both hosts. Each state variable contains two subscripts: one identifying the species (S-T. sanguisuga, G-T. gerstaeckeri, R-raccoon, W-woodrat) and the other representing the cell location (i represents the current cell, while n, s, e, w represents the 4 possible adjacent cells (north, south, east, and west) to the current cell. Each vector equation contains 8 migration terms, representing the bidirectional movement between cells assuming a von Neumann radius. Not every equation in every cell will contain all migration terms. Specific migration terms will be 0 if there is no corresponding vector species population in the specific adjacent cell or if the current cell is on a grid boundary. The parameter definitions and values are given in Table 11.

Table 11 Parameter definitions and values

Table 12 Values for sequence of lateral dispersal distances, {b _j } (in units m) and values for Ψ (units proportion/m²) from Crawford and Kribs-Zaleta (2013a). The generating function and derivation for Ψ is given in Crawford and Kribs-Zaleta (2013a)

$$\begin{aligned} \dot{S}_S[i,j] =&r_S \biggl(1- \frac{N_S[i,j]}{K_S[i,j]} \biggr)N_S[i,j]- \biggl(q_S \beta_R\frac{I_R[i,j]}{N_R[i,j]}+(1-q_S)\beta_{WS} \frac{I_W[i,j]}{N_W[i,j]} \biggr) \\&{}\times S_S[i,j]-\mu_S S_S[i,j] -(M_{sN}+M_{sE}+M_{sS}+M_{sW})S_S[i,j] \\&{}+M_{sN} S_S[i+1,j]+M_{sE} S_S[i,j-1]+M_{sS} S_S[i-1,j] \\&{}+M_{sW} S_S[i,j+1] \\\dot{I}_S[i,j] =& \biggl(q_S\beta_R \frac{I_R[i,j]}{N_R[i,j]}+(1-q_S)\beta _{WS}\frac{I_W[i,j]}{N_W[i,j]} \biggr)S_S[i,j]-\mu_S I_S[i,j] \\&{}-(M_{sN}+M_{sE}+M_{sS}+M_{sW})I_S[i,j]+M_{sN} I_S[i+1,j] \\&{}+M_{sE} I_S[i,j-1]+M_{sS} I_S[i+1,j]+M_{sW} I_S[i,j+1] \\\dot{S}_R[i,j] =&r_R \bigl(S_R[i,j]+(1-p_R)I_R[i,j] \bigr) \biggl(1-\frac {N_R[i,j]}{K_R[i,j]} \biggr)-\beta_{S2} \frac{I_S[i,j]}{N_R[i,j]}S_R[i,j] \\&{}-\mu_R S_R[i,j] \\\dot{I}_R[i,j] =&p_Rr_R I_R[i,j] \biggl(1-\frac{N_R[i,j]}{K_R[i,j]} \biggr)+\beta_{S2} \frac{I_S[i,j]}{N_R[i,j]}S_R[i,j]-\mu_R I_R[i,j] \\\dot{S}_G[i,j] =&r_G \biggl(1-\frac{N_G[i,j]}{K_G[i,j]} \biggr)N_G[i,j]-\beta _{W2}\frac{I_W[i,j]}{N_W[i,j]}S_G[i,j]- \mu_G S_G[i,j] \\&{}-(M_{gN}+M_{gE}+M_{gS}+M_{gW})S_G[i,j]+M_{gN} S_G[i+1,j] \\&{}+M_{gE} S_G[i,j-1]+M_{gS} S_G[i-1,j]+M_{gW} S_G[i,j+1] \\\dot{I}_G[i,j] =&\beta_{W2}\frac{I_W[i,j]}{N_W[i,j]}S_G[i,j]- \mu_G I_G[i,j] \\&{}-(M_{gN}+M_{gE}+M_{gS}+M_{gW})I_G[i,j] \\&{}+M_{gN} I\bigl(G[i+1,j]\bigr)+M_{gE} I_G[i,j-1]+M_{gS} I_G[i-1,j] \\&{}+M_{gW} I_G[i,j+1] \\\dot{S}_W[i,j] =&r_W \bigl(S_W[i,j]+(1-p_W)I_W[i,j] \bigr) \biggl(1-\frac {N_W[i,j]}{K_W[i,j]} \biggr) \\&{}- \biggl((1-q_W) \beta_G\frac{I_G[i,j]}{N_W[i,j]}+q_W\beta_{SW} \frac{I_S[i,j]}{N_W[i,j]} \biggr)S_W[i,j] \\&{}-\mu_W S_W[i,j] \\\dot{I}_W[i,j] =&p_W r_W I_W[i,j] \biggl(1-\frac{N_W[i,j]}{K_W[i,j]} \biggr)+ \biggl((1-q_W) \beta_G\frac{I_G[i,j]}{N_W[i,j]} \\&{}+q_W\beta_{SW} \frac {I_S[i,j]}{N_W[i,j]} \biggr)S_W[i,j] -\mu_W I_W[i,j] \end{aligned}$$

(1)