Spatiotemporal Dynamics of Virus Infection Spreading in Tissues

Abstract

Virus spreading in tissues is determined by virus transport, virus multiplication in host cells and the virus-induced immune response. Cytotoxic T cells remove infected cells with a rate determined by the infection level. The intensity of the immune response has a bell-shaped dependence on the concentration of virus, i.e., it increases at low and decays at high infection levels. A combination of these effects and a time delay in the immune response determine the development of virus infection in tissues like spleen or lymph nodes. The mathematical model described in this work consists of reaction-diffusion equations with a delay. It shows that the different regimes of infection spreading like the establishment of a low level infection, a high level infection or a transition between both are determined by the initial virus load and by the intensity of the immune response. The dynamics of the model solutions include simple and composed waves, and periodic and aperiodic oscillations. The results of analytical and numerical studies of the model provide a systematic basis for a quantitative understanding and interpretation of the determinants of the infection process in target organs and tissues from the image-derived data as well as of the spatiotemporal mechanisms of viral disease pathogenesis, and have direct implications for a biopsy-based medical testing of the chronic infection processes caused by viruses, e.g. HIV, HCV and HBV.

Fig 1. Schematic representation of virus infection dynamics regulation (left) and qualitative forms of the function f(v) (right).

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

Fig 2. Schematic representation of the spatial patterns of virus infection dynamics as travelling waves.

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

Fig 3. Wave propagation in the bistable case (left) for Eq (1) with function f(v) given by formula (23) (a = 1.1, b = 0.1, w₀ = 0.1, D = 10⁻⁴).

The curves show the function v(x, t) at successive moments of time. The wave speed dependence on time delay (right). The lower curve shows the results of numerical simulations, the upper curve is the analytical approximation by formula (34).

Fig 4. Propagating wave in the monostable case is monotone for small time delay (τ = 1, left) and non-monotone for large time delay (τ = 8, right).

The values of other parameters are a = 0.1, b = 0.3, w₀ = 0.1, D = 10⁻⁴.

Fig 5. Numerical simulations of eq (1) with the function f(v) = rv (r = 2, D = 10⁻⁴).

Wave propagation for three different values of time delay, τ = 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.

Fig 6. Numerical simulations of different regimes of infection spreading depending on time delay, τ = 0.4, 0.95, 1.5, 10;D = 0.0001.

For small time delay (two left figures: τ = 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: τ = 1.5, 10), the second wave propagates faster and they finally merge forming a single wave which can be either monotone or non-monotone.

Fig 7. Spatiotemporal regimes of infection spreading.

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 v₁. Space oscillations become more complex for larger values of τ 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, τ = 0.7, 1, 1.5, 2;D = 0.0001.

Fig 8. Existence of a monostable wave with spatial oscillations behind it.

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 c₀. The values of time delay are, respectively, τ = 1, 2, 3, 4;D = 0.0001.

Fig 9. Different regimes of infection wave propagation on the parameter plane (f_m is the maximum of the function f(v), and τ is the time needed for the development of immune response), see Supplementary materials: 1—stationary wave propagation (cf. Fig 6, two right images, Fig 7, right image), 2—two consecutive waves with different speeds (cf. Fig 6, two left images), 3—two waves with spatiotemporal patterns between them (cf. Fig 7, three left images and Fig 8).