On the Spread of a Virus and Traveling Waves, Part I
Abstract
This paper develops a one-dimensional model for a mobile viral component propagating through a spatially distributed susceptible population. Random viral motion produces diffusion, whereas a response to the relative gradient of susceptible density generates directed transport. The existence of a positive entire traveling wave is proved when directed transport dominates diffusion, and the limiting case of equal transport and diffusion coefficients is constructed separately. A dimensionless transformation then converts the remaining equation into a logistic equation and supplies closed-form profiles. These formulas identify the unique viral maximum and the far-field behavior. A conservative implicit-explicit finite-difference method provides an independent numerical check. Diffusion is evaluated at the new time level, the gradient-driven flux explicitly at cell interfaces, and susceptible depletion pointwise. Exact traveling-wave values specify the initial data and time-dependent Dirichlet conditions at both endpoints. Thus, the computation tests preservation and translation of the analytical profiles. A calculation with eight hundred spatial subintervals reproduces the localized viral band and the monotone susceptible front. The largest uniform errors are approximately 1.70·10−2 for viral density and 2.22·10−3 for susceptible density. The numerical results corroborate the analytical construction and show that the conservative discretization captures the proposed propagation mechanism.
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BibTeX
@misc{wang2026spread,
title = {{On the Spread of a Virus and Traveling Waves, Part I}},
author = {Yizhou Wang and Akif Ibragimov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04732},
eprint = {2610.04732},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04732},
}
RIS
TY - UNPB
TI - On the Spread of a Virus and Traveling Waves, Part I
AU - Yizhou Wang
AU - Akif Ibragimov
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04732
UR - https://arxiv.org/abs/2610.04732
AB - This paper develops a one-dimensional model for a mobile viral component propagating through a spatially distributed susceptible population. Random viral motion produces diffusion, whereas a response to the relative gradient of susceptible density generates directed transport. The existence of a positive entire traveling wave is proved when directed transport dominates diffusion, and the limiting case of equal transport and diffusion coefficients is constructed separately. A dimensionless transformation then converts the remaining equation into a logistic equation and supplies closed-form profiles. These formulas identify the unique viral maximum and the far-field behavior. A conservative implicit-explicit finite-difference method provides an independent numerical check. Diffusion is evaluated at the new time level, the gradient-driven flux explicitly at cell interfaces, and susceptible depletion pointwise. Exact traveling-wave values specify the initial data and time-dependent Dirichlet conditions at both endpoints. Thus, the computation tests preservation and translation of the analytical profiles. A calculation with eight hundred spatial subintervals reproduces the localized viral band and the monotone susceptible front. The largest uniform errors are approximately 1.70·10^−2 for viral density and 2.22·10^−3 for susceptible density. The numerical results corroborate the analytical construction and show that the conservative discretization captures the proposed propagation mechanism.
ER -