Analysis of Transmission of Infection in Epidemics: Confined Random Walkers in Dimensions Higher Than One

Abstract

The process of transmission of infection in epidemics is analyzed by studying a pair of random walkers, the motion of each of which in two dimensions is confined spatially by the action of a quadratic potential centered at different locations for the two walks. The walkers are animals such as rodents in considerations of the Hantavirus epidemic, infected or susceptible. In this reaction–diffusion study, the reaction is the transmission of infection, and the confining potential represents the tendency of the animals to stay in the neighborhood of their home range centers. Calculations are based on a recently developed formalism (Kenkre and Sugaya in Bull Math Biol 76:3016–3027, 2014) structured around analytic solutions of a Smoluchowski equation and one of its aims is the resolution of peculiar but well-known problems of reaction–diffusion theory in two dimensions. The resolution is essential to a realistic application to field observations because the terrain over which the rodents move is best represented as a 2-d landscape. In the present analysis, reaction occurs not at points but in spatial regions of dimensions larger than 0. The analysis uncovers interesting nonintuitive phenomena one of which is similar to that encountered in the one-dimensional analysis given in the quoted article, and another specific to the fact that the reaction region is spatially extended. The analysis additionally provides a realistic description of observations on animals transmitting infection while moving on what is effectively a two-dimensional landscape. Along with the general formalism and explicit one-dimensional analysis given in Kenkre and Sugaya (2014), the present work forms a model calculational tool for the analysis for the transmission of infection in dilute systems.

Appendix: Analysis in One Dimension: Introduction of Infection Range

We consider that the infection is transmitted at a range b, while the mice move in one dimension, and start the analysis in CM-relative coordinates. The equation of motion for $P(x_{+},x_{-},t)$, the probability density to find the walkers at CM and relative coordinates $x_+$ and $x_-$, respectively, at time t is

$$\begin{aligned} \frac{\partial P(x_+,x_-,t)}{\partial t}&= \frac{\partial }{\partial x_{+}}\gamma \left( x_{+}-h_{+}\right) P(x_+,x_-,t)+\frac{\partial }{\partial x_{-}}\gamma \left( x_{-}-h_{-}\right) P(x_+,x_-,t) \nonumber \\&\quad +\,D\left( \frac{\partial ^2}{\partial x_{+}^2}+\frac{\partial ^2}{\partial x_{-}^2}\right) P(x_+,x_-,t)\nonumber \\&\quad -\,\mathcal {C}_1\delta \left( x_{-}-b\right) P(x_{+},x_{-},t)-\mathcal {C}_{1}\delta \left( x_{-}+b\right) P(x_{+},x_{-},t). \end{aligned}$$

(A.1)

The first three terms represent the Smoluchowski motion. The infection transmission is described in the last two terms; the arguments of the $\delta$-functions indicate that the infection is transmitted to the susceptible mouse from the infected one when they are at distance b apart, i.e., when $x_{-}=\pm b$. The infection rate is given by $\mathcal {C}_{1}$.

The symmetry and simplicity of this infection region allows for an exact calculation of $\tilde{\nu }(\epsilon )$, without the approximation of the $\nu$-function method. The propagator for this problem is given by

$$\begin{aligned} \varPi (x_{+},x'_{+},x_{-},x'_{-},t) = \frac{1}{4\pi D\mathcal {T}(t)}e^{-\frac{\left( x_{+}-h_{+}-(x'_{+}-h_{+})e^{-\gamma t}\right) ^2+\left( x_{-}-h_{-}-(x'_{-}-h_{-})e^{-\gamma t}\right) ^2}{4D\mathcal {T}(t)}}. \end{aligned}$$

(A.2)

In terms of this propagator, the solution to Eq. (A.1) in the Laplace domain as

$$\begin{aligned} \tilde{P}(x_{+},x_{-},\epsilon )&= \tilde{\eta }(x_{+},x_{-},\epsilon ) \nonumber \\&\quad -\mathcal {C}_{1}\int _{-\infty }^{\infty }dx'_{+}\tilde{\varPi }(x_{+},x'_{+},x_{-},-b,\epsilon )\tilde{P}(x'_{+},-b,\epsilon ) \nonumber \\&\quad -\mathcal {C}_{1}\int _{-\infty }^{\infty }dx'_{+}\tilde{\varPi }(x_{+},x'_{+},x_{-},b,\epsilon )\tilde{P}(x'_{+},b,\epsilon ). \end{aligned}$$

(A.3)

From its definition given in Eq. (3), the infection probability in the Laplace domain is given in Eq. (A.3) by

$$\begin{aligned} \widetilde{\mathcal {I}}(\epsilon ) = \frac{\mathcal {C}_{1}}{\epsilon }\left[ \int _{-\infty }^{\infty }dx_{+}\widetilde{P}(x_{+},b,\epsilon )+\int _{-\infty }^{\infty }dx_{+}\widetilde{P}(x_{+},-b,\epsilon )\right] . \end{aligned}$$

(A.4)

The use of the defect technique, i.e., setting $x_{-}=\pm b$ and integrating $x_{+}$ over all space, after some algebra, yields

$$\begin{aligned}&\int _{-\infty }^{\infty }dx_{+}\widetilde{P}(x_{+},b,\epsilon )+\int _{-\infty }^{\infty }dx_{+}\widetilde{P}(x_{+},-b,\epsilon )\nonumber \\&\quad = \frac{1}{\mathcal {C}_{1}}\frac{\left[ 1/\mathcal {C}_{1}+\tilde{\nu }^{++}_{1}(\epsilon )-\tilde{\nu }^{+-}_{1}(\epsilon )\right] \tilde{\mu }^{-}_{1}(\epsilon )+\left[ 1/\mathcal {C}_{1}+\tilde{\nu }^{--}_{1}(\epsilon )-\tilde{\nu }^{-+}_{1}(\epsilon )\right] \tilde{\mu }^{+}_{1}(\epsilon )}{\left[ 1/\mathcal {C}_{1}+\tilde{\nu }^{--}_{1}(\epsilon )\right] \left[ 1/\mathcal {C}_{1}+\tilde{\nu }^{++}_{1}(\epsilon )\right] -\tilde{\nu }^{-+}_{1}(\epsilon )\tilde{\nu }^{+-}_{1}(\epsilon )}, \end{aligned}$$

(A.5)

where

$$\begin{aligned} \mu ^{+}_{1}(t)&=\int _{-\infty }^{\infty }dx_{+}\eta (x_{+},b,t) = \frac{1}{\sqrt{4\pi D\mathcal {T}(t)}}\,e^{-\frac{\left( b-H-(x_{-}^{0}-H)e^{-\gamma t}\right) ^2}{4D\mathcal {T}(t)}} \end{aligned}$$

(A.6)

$$\begin{aligned} \mu ^{-}_{1}(t)&=\int _{-\infty }^{\infty }dx_{+}\eta (x_{+},-b,t) = \frac{1}{\sqrt{4\pi D\mathcal {T}(t)}}\,e^{-\frac{\left( b+H+(x_{-}^{0}-H)e^{-\gamma t}\right) ^2}{4D\mathcal {T}(t)}} \end{aligned}$$

(A.7)

$$\begin{aligned} \nu ^{++}_{1}(t)&= \int _{-\infty }^{\infty }dx_{+}\varPi (x_{+},b,x'_{+},b)= \frac{1}{\sqrt{4\pi D\mathcal {T}(t)}}\,e^{-\frac{\left[ (b-H)(1-e^{-\gamma t})\right] ^2}{4D\mathcal {T}(t)}} \end{aligned}$$

(A.8)

$$\begin{aligned} \nu ^{+-}_{1}(t)&= \int _{-\infty }^{\infty }dx_{+}\varPi (x_{+},b,x'_{+},-b)=\frac{1}{\sqrt{4\pi D\mathcal {T}(t)}}\,e^{-\frac{\left[ b-H+(b+H)e^{-\gamma t}\right] ^2}{4D\mathcal {T}(t)}} \end{aligned}$$

(A.9)

$$\begin{aligned} \nu ^{-+}_{1}(t)&=\int _{-\infty }^{\infty }dx_{+}\varPi (x_{+},-b,x'_{+},b)= \frac{1}{\sqrt{4\pi D\mathcal {T}(t)}}\,e^{-\frac{\left[ b+H+(b-H)e^{-\gamma t}\right] ^2}{4D\mathcal {T}(t)}} \end{aligned}$$

(A.10)

$$\begin{aligned} \nu ^{--}_{1}(t)&=\int _{-\infty }^{\infty }dx_{+}\varPi (x_{+},-b,x'_{+},-b)= \frac{1}{\sqrt{4\pi D\mathcal {T}(t)}}\,e^{-\frac{\left[ (b+H)(1-e^{-\gamma t})\right] ^2}{4D\mathcal {T}(t)}}. \end{aligned}$$

(A.11)

In the calculation of the $\mu _{1}(t)$’s, a $\delta$-function initial condition was assumed. The infection probability in the Laplace domain is then given exactly by

$$\begin{aligned} \widetilde{\mathcal {I}}(\epsilon ) = \frac{1}{\epsilon }\frac{\left[ 1/\mathcal {C}_{1}+\tilde{\nu }^{++}_{1}(\epsilon )-\tilde{\nu }^{+-}_{1}(\epsilon )\right] \tilde{\mu }^{-}_{1}(\epsilon )+\left[ 1/\mathcal {C}_{1}+\tilde{\nu }^{--}_{1}(\epsilon )-\tilde{\nu }^{-+}_{1}(\epsilon )\right] \tilde{\mu }^{+}_{1}(\epsilon )}{\left[ 1/\mathcal {C}_{1}+\tilde{\nu }^{--}_{1}(\epsilon )\right] \left[ 1/\mathcal {C}_{1}+\tilde{\nu }^{++}_{1}(\epsilon )\right] -\tilde{\nu }^{-+}_{1}(\epsilon )\tilde{\nu }^{+-}_{1}(\epsilon )}. \end{aligned}$$

(A.12)

The non-monotonic effect is explained in detail in Ref. (Kenkre and Sugaya 2014) in the steady state and in the contact limit where $1/\mathcal {C}_{1}$ is much greater than any of the $\nu _{1}(t)$’s and $\mu _{1}(t)$’s in its effect. In these limit, $\mathcal {I}(t)$ approximately becomes

$$\begin{aligned} \mathcal {I}(t) \sim \mathcal {C}_{1}\left( \mu ^{+}_{1}(\infty )+\mu ^{-}_{1}(\infty )\right) \cdot t, \end{aligned}$$

(A.13)

where we note that $\nu ^{++}_{1}(\infty )=\nu ^{+-}_{1}(\infty )=\mu ^{+}_{1}(\infty )$ and $\nu ^{--}_{1}(\infty )=\nu ^{-+}_{1}(\infty )=\mu ^{-}_{1}(\infty )$. The condition for the optimal value of $\gamma \tau _H$ value found from this result yields the transcendental relation,

$$\begin{aligned} \frac{1-2\gamma \tau _H(1-b/H)^2}{1-2\gamma \tau _H(1+b/H)^2} = -e^{-4\gamma \tau _H(b/H)}. \end{aligned}$$

(A.14)

The value of $\gamma \tau _H$ for given value of b / H found from this equation is plotted in Fig. 4.