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.