A multiscale model of virus pandemic: Heterogeneous interactive entities in a globally connected world

Abstract

This paper is devoted to the multidisciplinary modelling of a pandemic initiated by an aggressive virus, specifically the so-called SARS-CoV-2 Severe Acute Respiratory Syndrome, corona virus n.2. The study is developed within a multiscale framework accounting for the interaction of different spatial scales, from the small scale of the virus itself and cells, to the large scale of individuals and further up to the collective behaviour of populations. An interdisciplinary vision is developed thanks to the contributions of epidemiologists, immunologists and economists as well as those of mathematical modellers. The first part of the contents is devoted to understanding the complex features of the system and to the design of a modelling rationale. The modelling approach is treated in the second part of the paper by showing both how the virus propagates into infected individuals, successfully and not successfully recovered, and also the spatial patterns, which are subsequently studied by kinetic and lattice models. The third part reports the contribution of research in the fields of virology, epidemiology, immune competition, and economy focussed also on social behaviours. Finally, a critical analysis is proposed looking ahead to research perspectives.

Keywords: 92C60; 92D30; COVID-19; SARS-CoV-2; complexity; immune competition; intracellular infection dynamics; living systems; multiscale problems; networks; spatial patterns; viral quasispecies; virus structure modelling.

Fig. 1.

A crowd with aggregation of multiple groups.

Fig. 2.

Transfer diagram of the model. Boxes represent functional subsystems and arrows indicate transition of individuals.

Fig. 3.

Dynamics within infected population.

Fig. 4.

(Color online) α = 0.4. Left: Total number of active cases (blue), active cases requiring hospitalisation (red) and the number of available beds (black). Right: Cumulative infected population (red), recovered (blue) and dead (black) versus time.

Fig. 5.

(Color online) α = 0.3. Left: Total number of active cases (blue), active cases requiring hospitalisation (red) and the number of available beds (black). Right: Cumulative infected population (red), recovered (blue) and dead (black) versus time.

Fig. 6.

(Color online) α = 0.25. Left: Total number of active cases (blue), active cases requiring hospitalisation (red) and the number of available beds (black). Right: Cumulative infected population (red), recovered (blue) and dead (black) versus time.

Fig. 7.

Left: Proportion of people infected at the same time when the peak of infection is reached as a function of α. Right: Time at which this peak occurs as function of α.

Fig. 8.. α

α = 0.4 for t < 300, then reduced to α = 0.25 at t = 300, and set back to the “normal/initial” state at t = 1200. Left: Total number of active cases. Right: Estimated number of patients requiring ICU admission in relation to system capacity.

Fig. 9.. α

α = 0.4 for t < 300, then reduced to α = 0.25 at t = 300, and set back to the “normal/initial” state at t = 900. Left: Total number of active cases. Right: Estimated number of patients requiring ICU admission in relation to system capacity.

Fig. 10.

(Color online) Healthy and cumulative infected populations for different values of u_c, namely, c = 5 (yellow), c = 4 (red), c = 3 (blue), by simulations with m = 5.

Fig. 11.

A crowd during city traffic.

Fig. 12.

The economic damage as a function of the lockdown.

Fig. 13.

The economic damage function across income levels.

Fig. 14.

Transfer diagram of the model. Boxes represent functional subsystems and arrows indicate transition of individuals.

Fig. 15.

(Color online) Sensitivity to κ. (a) Blue: α = 0.4, γ = 0.2, Red: α = 0.2, γ = 0.1, Yellow: α = 0.1, γ = 0.05. (b) Blue: β = 0.15, γ = 0.3, Red: β = 0.1, γ = 0.2, Yellow: β = 0.05, γ = 0.1. (c) Blue: α = 0.4, γ = 0.2, Red: α = 0.2, γ = 0.1, Yellow: α = 0.1, γ = 0.05. (d) Blue: β = 0.15, γ = 0.3, Red: β = 0.1, γ = 0.2, Yellow: β = 0.05, γ = 0.1.

Fig. 16.

Varying locking times. We take T_ℓ = 100, 200, 300, and fixed lock-open time T_d = 1200. α = 0.4 for t ∈ [0, T_ℓ) ∪ [T_d, T_max] while α = 0.25 during the locking interval.

Fig. 17.

Varying de-locking times. We take a fixed locking time T_ℓ = 300, and three different lock-open times T_d = 900, 1200,1500. α = 0.4 for t ∈ [0, T_l) ∪ [T_d, T_max] while α = 0.25 during the locking interval. Note that the three curves coincide until the first lock-open time, and are consequently represented in black in that interval.

Fig. 18.

Varying the de-locking value α_d. We take fixed locking and lock-open times T_l = 300 and T_d = 1200, respectively. α = 0.4 initially for t ∈ [0, T_l) then reduced to α = 0.25 during the locking interval and finally we consider three different lock-open values α_d = 0.3, 0.4, 0.5. Note that the three curves coincide until the lock-open time, and are consequently represented in black in that interval.