# Kinetics of epidemic spread in a low-mobile population

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The standard method of including the spatial effects into epidemiological problems consists in the introduction of a of diffusion term leading to solutions in form of traveling waves (Fisher-Kolmogorov waves).  However, from the point of view of the mechanism of infection propagation in a low-mobility population the diffusion term (corresponding to a random walk) borrowed from the theory of diffusion-controlled reactions, looks a bit an artificial addition. It is especially clear in the case of plant diseases – plants do not move at all, or animals showing a high degree of site fidelity (like harbor seals: the mixing within the population located at one site is strong while the infection transmission between sites takes place only via contacts between individuals performing rare relatively long trips between neighboring sites). Nevertheless the infection waves appear also in this situation.

In what follows, we consider infection wave propagation in a population of low-mobile individuals, where a local infection spread mechanism (between individuals or parts of the population in close contact) is present.

Such a mechanism corresponds to what is known as a ‘contact process’ in statistical physics. A population consisting of three kinds of individuals, namely the susceptible (S), the infected (I), and the recovered/removed (R) is subdivided into cells with strong mixing within a population located in one site (cell) and the infection transmission between sites only via contacts along boundaries of neighbor cells.

\begin{eqnarray*}
\Delta I(x, y)=\kappa S(x, y)\times\\
\left( \frac{1}{4}I(x+a, y)+\right.\\
\frac{1}{4}I(x-a, y)+\\
\frac{1}{4}I(x, y+a)+\\
\left.\frac{1}{4}I(x, y-a)\right) \Delta t
\end{eqnarray*}

Analytical continuous approximation:
\begin{eqnarray*}
\partial_t I &=&\kappa S\left( I+(a^{2}/4)\nabla
^{2}I\right)-R, \label{PDE} \\
\partial_t R &=&\tau^{-1}I, \\\nonumber
1&=&S+I+R
\end{eqnarray*}
can be exactly represented within a comoving frame $x’=x-vt$, $v=2\sqrt{D(\kappa-\tau^{-1})}$ as
\begin{eqnarray*}
v\frac{dS}{dx’}&=&\kappa S(1-S)-DS\frac{d^2S}{dx’^2} +\frac{1}{\tau}S\ln S, \\
-\ln S&=&\kappa\tau R+ D\tau\frac{d^2 R}{dx’^2}=0.
\end{eqnarray*}

The model is tested by comparison with the real dynamics of the epidemic of phocine distemper virus infection among harbor seals habitating Danish Straits in 1988.

The points of the first record of disease marked with stars and the points of 50% mortality marked with circles. The slope of the dashed line drawn through the arithmetic mean of each pair of experimental points (dots) except the initial one allows us to determine the mean velocity of infection as approximately 1.43 km/day that shows a good agreement with the calculated via the proposed model value 1.47 km/day.

Details can be found in the articles:
• E.B. Postnikov, I.M. Sokolov. Continuum description of a contact infection spread in a SIR model. Mathematical Biosciences. 2007. V. 208. Iss.1. pp. 205–215.
• U. Naether, E.B. Postnikov, I.M. Sokolov. Infection fronts in contact disease spread. The European Physical Journal B. 2008. V. 65. pp. 353-359.