Existence of a monostable wave with spatial oscillations behind it.

<p>This wave is separated from the bistable wave by a zone of irregular oscillations. Increase of the delay value results in a qualitative change of the spatial patterns of the infection spread. The two travelling waves do not merge and the monostable wave is not followed by steady space oscillations. Aperiodic oscillations are observed behind the wave front which propagates at a speed <i>c</i>₀. The values of time delay are, respectively, <i>τ</i> = 1, 2, 3, 4;<i>D</i> = 0.0001.</p

Spatiotemporal regimes of infection spreading.

<p>The monostable wave becomes non-monotone with decaying or persisting oscillations behind it. The type of patterns on the left is characterized by a transition zone between decaying space oscillations and the bistable wave with perturbed time oscillations of the homogeneous solution <i>v</i>₁. Space oscillations become more complex for larger values of <i>τ</i> which represent a second type of the spatial dynamics. If time delay is sufficiently large, then the two travelling waves merge, as before, forming a single stable non-monotone wave. The values of time delay are, respectively, <i>τ</i> = 0.7, 1, 1.5, 2;<i>D</i> = 0.0001.</p

Propagating wave in the monostable case is monotone for small time delay (<i>τ</i> = 1, left) and non-monotone for large time delay (<i>τ</i> = 8, right).

<p>The values of other parameters are <i>a</i> = 0.1, <i>b</i> = 0.3, <i>w</i>₀ = 0.1, <i>D</i> = 10⁻⁴.</p

Numerical simulations of different regimes of infection spreading depending on time delay, <i>τ</i> = 0.4, 0.95, 1.5, 10;<i>D</i> = 0.0001.

<p>For small time delay (two left figures: <i>τ</i> = 0.4, 0.95), there are two consecutive waves of infection propagating with different speeds. The first wave can be non-monotone. For large time delay (two right figures: <i>τ</i> = 1.5, 10), the second wave propagates faster and they finally merge forming a single wave which can be either monotone or non-monotone.</p

Schematic representation of the spatial patterns of virus infection dynamics as travelling waves.

<p>Typical wavefront solutions taking a steady state value <i>v</i>_<i>i</i> at the left end and another steady state <i>v</i>_<i>j</i>, <i>i</i> ≠ <i>j</i> at the right end. The travelling waves evolve with the speed <i>c</i>. A qualitative relationship between the initial viral load and the emerging pattern on the spatiotemporal pattern of virus spread is sketched.</p

Schematic representation of virus infection dynamics regulation (left) and qualitative forms of the function <i>f</i>(<i>v</i>) (right).

<p>Low level infection stimulates immune response while high level infection down-regulates it. The former corresponds to the growing branch of the function <i>f</i>(<i>v</i>) while the latter to its decreasing branch.</p

Numerical simulations of eq (1) with the function <i>f</i>(<i>v</i>) = <i>rv</i> (<i>r</i> = 2, <i>D</i> = 10⁻⁴).

<p>Wave propagation for three different values of time delay, <i>τ</i> = 1.4, 2, 4, respectively. For small time delay (left) space and time oscillations decay, for intermediate time delay (middle) space oscillations decay while time oscillations persists, for large time delay (right) both of them persist.</p