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This paper models a one-dimensional mobile viral component's propagation through a susceptible population, distinguishing between diffusive random motion and directed transport driven by susceptible density gradients. It proves the existence of positive entire traveling waves when directed transport exceeds diffusion, constructing a separate solution for the equal transport and diffusion case. Closed-form profiles are derived via a dimensionless transformation, identifying viral maxima and far-field behavior, and are numerically validated by a conservative implicit-explicit finite-difference method.
Directed transport can drive traveling wave solutions in viral spread models, even outperforming diffusion.
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 $1.70\cdot10^{-2}$ for viral density and $2.22\cdot10^{-3}$ for susceptible density. The numerical results corroborate the analytical construction and show that the conservative discretization captures the proposed propagation mechanism.