Complexity and anisotropy in host morphology make populations less susceptible to epidemic outbreaks

Abstract

One of the challenges in epidemiology is to account for the complex morphological structure of hosts such as plant roots, crop fields, farms, cells, animal habitats and social networks, when the transmission of infection occurs between contiguous hosts. Morphological complexity brings an inherent heterogeneity in populations and affects the dynamics of pathogen spread in such systems. We have analysed the influence of realistically complex host morphology on the threshold for invasion and epidemic outbreak in an SIR (susceptible-infected-recovered) epidemiological model. We show that disorder expressed in the host morphology and anisotropy reduces the probability of epidemic outbreak and thus makes the system more resistant to epidemic outbreaks. We obtain general analytical estimates for minimally safe bounds for an invasion threshold and then illustrate their validity by considering an example of host data for branching hosts (salamander retinal ganglion cells). Several spatial arrangements of hosts with different degrees of heterogeneity have been considered in order to separately analyse the role of shape complexity and anisotropy in the host population. The estimates for invasion threshold are linked to morphological characteristics of the hosts that can be used for determining the threshold for invasion in practical applications.

Figure 1.

Example of a system formed by complex hosts represented, for concreteness, by planar neurons corresponding to salamander retinal ganglion cells placed on the nodes of a triangular lattice with lattice spacing a in such a way that the somata coincide with the lattice nodes. The pathogen infests the surroundings (shaded area) of infected (I) hosts and, eventually, reaches the neighbouring susceptible hosts (e.g. amber susceptible (S) neuron on the right from I). The probability of infection of a susceptible host and the probability of a global epidemic outbreak depend on overlaps, J, between the infested region and susceptible hosts, and infection efficiency, k. The overlaps are dictated by the host morphology and spatial arrangement of hosts. The infection efficiency determines the effectiveness of the contact in terms of transmission of infection. Besides these factors, the inset shows that J and thus the invasion threshold depend, in general, on the local orientation of the hosts, ϕ.

Figure 2.

Probability of invasion, P_inv, for morphologically complex hosts in disordered arrangements of type 1 on a lattice of size L × L = 200 × 200. The main plot displays P_inv as a function of the average transmissibility, , for heterogeneous systems with different lattice spacings a (marked by different symbols) and for a mean-field system with homogeneous transmissibility (dashed line). The bond-percolation critical probability, p_c, marked by the arrow gives the invasion threshold in the thermodynamic limit. The inset shows the invasion probability as a function of the infection efficiency k for different values of lattice spacing marked by the same symbols as in the main figure. Circle, a = 20; asterisk, a = 40; square, a = 60; triangle, a = 80; diamond, a = 100.

Figure 3.

Invasion threshold for the system of branching hosts. (a) Representation in terms of the infection efficiency, k, and the lattice spacing, a. The threshold k_c(a) for arrangements of type 1 is shown by circles. The shaded region corresponds to the statistically possible values for the estimation of k_c(a) in terms of effective circular hosts with homogeneous overlaps defined in equation (3.10). Diamonds indicate the average thresholds (solid symbols) and (open symbols) for arrangements of type 3. (b) Invasion threshold k_c as a function of the average overlap for arrangements of type 1 (circles), type 2 with orientation ϕ = 0 (diamonds) and type 3 with mean orientation and two widths of uniform distribution, (solid squares) and (open squares). The branching host used as a motif for arrangements of type 2 and 3 is displayed in the figure. The dashed line represents the dependence of k_c⁰ versus given by equation (3.2). The insets show the dispersions V₁ and V₂ of the overlaps associated with the disorder and anisotropy, respectively, corresponding to the same arrangements as in the main figure (the symbol code is the same as in the main figure).

Figure 4.

Arrangements of type 2 (an identical host with the same orientation placed at all the nodes). (a) Unit cell of two different configurations constructed by using the same branching host. The orientation, ϕ, and spacing, a, corresponding to each case are indicated in the schematic reference frames displayed at the top. As shown in the left frame, there are three different values of the overlaps, {J₁, J₂, J₃}, corresponding to each of the main directions in the lattice. (b) Space of overlaps, (J₁, J₂, J₃). Each configuration with different a and ϕ is mapped into a point in this space. The set of overlaps corresponding to all the possible configurations obtained for a given host defines its overlap locus (surface in blue). The deviation of the blue surface from the straight black line representing overlaps between isotropic hosts (J₁ = J₂ = J₃) shows the degree of anisotropy in the overlaps for a typical branching host (shown in (a)). High values of the overlaps correspond to small lattice spacings a. In particular, the points labelled as C1 and C2 correspond to the configurations shown in (a). The critical surface defined by g(J₁, J₂, J₃, k) = 0 (see equation (3.5)) is shown in red for two values of the infection efficiency: k = 0.02 and k = 0.008. For a given value of k, the points in the overlap locus below/above the critical surface correspond to safe/vulnerable configurations. The intersection of the overlap locus with the critical surfaces parametrized by k determines the critical threshold . The critical infection efficiency corresponding to a configuration with lattice spacing a and orientation ϕ is given by the value of k related to the critical surface containing the configuration point .

Figure 5.

Dependence of the critical lattice spacing, , on the SLAD, d₂, for all the branching hosts in arrangements of type 2. Different symbols refer to different values of k and each point for a particular symbol corresponds to an individual branching host. The solid lines represent the linear regression fit for each value of k (e.g. a_c(k) = 14.1 + 0.94d₂ with correlation coefficient ≃ 0.97 for k = 1). The inset defines graphically the LADs {d₁, d₂, d₃}. Circle, k = 0.1; square, k = 1; triangle, k = 10.