Extinction of refugia of hantavirus infection in a spatially heterogeneous environment

We predict an abrupt observable transition, on the basis of numerical studies, of hantavirus infection in terrain characterized by spatially dependent environmental resources. The underlying framework of the analysis is that of Fisher equations with an internal degree of freedom, the state of infection. The unexpected prediction is of the sudden disappearance of refugia of infection in spite of the existence of supercritical (favorable) food resources, brought about by reduction of their spatial extent. Numerical results are presented and a theoretical explanation is provided on analytic grounds on the basis of the competition of diffusion of rodents carrying the hantavirus and nonlinearity present in the resource interactions.

Figure 1

(Color online) Left: Landsat satellite image of the Jemez Mountains in northern New Mexico during winter, illustrating the isolated nature of forested mountain ranges in the southwestern United States. The Rio Grande valley is to the right of the image, running from north (top) to south (bottom). Right: Habitats transition along an elevation gradient within the Jemez Mountains (Valles Caldera National Preserve). Deer mouse and hantavirus refugia are typically found within stands of Ponderosa pine/oak and mixed conifer forests, but in high productivity years, mouse populations will increase and expand in spatial extent; in low productivity years, mouse populations shrink in size and distribution, potentially leading to an extinction of the hantavirus infection within an isolated deer mouse population.

Figure 2

Step-shaped environmental parameter $K$ with supercritical value $K=K_p$ within the region of width $L$ and subcritical value $K=K_b$ outside the region as shown, the critical value $K_c$ lying between $K_p$ and $K_b$. The inset shows the steady state infected mouse density $M_I$ obtained numerically by solving Eq. (1) displayed as a log-linear plot: the abscissa in the inset is the distance $x$ (schematic) while the ordinate is the logarithm of $M_I$. The straight lines away from the supercritical region show that $M_I$ decays essentially in an exponential manner. Parameters used are arbitrary. In appropriate units, $a=0.1,b=1.0,c=0.5,D=20,K_p=30,K_b=10,K_c=20$.

Figure 3

Abrupt transition in the infection as the extent $L$ of the refugium is varied. Steady state mouse density profiles are shown for different values of $L$, the infected density $M_I$ in (a) and the susceptible density $M_S$ in (b) respectively. The infected density decreases with a decrease in $L$ ($L=100$, 40, 20 and 18.4 for solid, dashed, dotted and dashed-dotted) and vanishes completely at the critical value, $L=18.2$ in the units employed. The susceptible density, on the other hand, does not vanish even for $L=0$ ($L=100$, 40, 20, 16, and 0 for solid, long-dashed, dotted, dashed-dotted and small-dashed). Values of the densities (infected, susceptible and total) at the center of supercritical region are plotted in (c) as a function of $L$. Only the infected density exhibits a transition. Parameters are, in arbitrary units: $a=0.1,b=1.0,c=0.5,D=20$.

Figure 4

The critical length $L_c$ plotted for (arbitrary) parameters $a=0.1,b=1.0,c=0.5,K_b=10$ showing power law dependence on (a) the diffusion coefficient $D$ where the plot is on a log-log scale (base 10) and the straight line with slope 1/2 shows the square root power law dependence $L_c\sim \sqrt{D}$, the value of $K_p$ being 30, and on (b) the environmental parameter $K_p$ for the same parameters and $D=20$. Although the main plot in (b) is on a linear scale and shows a complex dependence on $K_p$, the inset is on a log-log scale (base 10) and against $K_p-K_c$ and the straight line with slope $-1/2$ shows the power law dependence $L_c\sim 1/\sqrt{K_p-K_c}$.

Figure 5

Validity of the linearization procedure used to obtain the steady state spatial density profile of the total mouse density $M\left(x\right)$. Displayed is $M\left(x\right)$ expressed in units of $K_b\left(b-c\right)$, $x$ being plotted in units of $1/\alpha =\sqrt{D/\left(b-c\right)}$. The solid lines are numerically obtained exact solutions of the full nonlinear equation while the dashed lines are the analytically obtained approximation of the linearized counterpart. The three plots, (a), (b), and (c) correspond to three different values of $\alpha L$; in (a) it equals 1.58, in (b) it is twice as large as in (a), and in (c) it is ten times as large as in (b). The value of $K_p/K_b$ is 3 in all plots. The procedure clearly becomes more accurate as $\alpha L$ increases.

Figure 6

Schematic representation of the proposed approximation approach for finding the critical length. Plotted is the spatial dependence of $\mathcal{A}\left(x\right)$ as obtained from the calculated $M\left(x\right)$ (solid line), and in its step profile representation with the constant values ${\mathcal{A}}_0$ inside and $-{\mathcal{A}}_1$ outside the refugium (dashed line). The result is the determination of $L_{eff}$ from the actual $L$.

Figure 7

(Color online) Comparison of our analytic theory with numerically obtained values for the dimensionless extinction length $\alpha L_c$ as a function of the quantities $K_p^{\ast }$, $K_b^{\ast }$ and $K_c^{\ast }$ in (a), (b), and (c), respectively. In (a), $K_c^{\ast }=2,K_b^{\ast }=1$, in (b) $K_p^{\ast }=3,K_c^{\ast }=2$ and in (c) $K_p^{\ast }=3,K_b^{\ast }=1$. The filled circles are obtained numerically from the full nonlinear time-dependent Eq. (1) and the black solid line is the analytic prediction of our theory as given by Eq. (15). In each of the three figures we also display, as dotted line, the prediction of the simple Eq. (16). That simplified theory qualitatively reproduces the correct behavior and is valid for small diffusion.