A Continuous Model of Three Scenarios of the Infection Process with Delayed Immune Response Factors

Abstract

The course of an infection was modeled as a controlled nonlinear process. Understanding the substantial differences observed in the trajectory of the disease caused by the new coronavirus SARS-CoV-2 is of critical importance at the moment. Numerous factors have been considered to explain the fact that symptoms vary highly among different people and the infection transmission rate varies among local populations. Virus replication within the host cell and the development of an immune response to virus antigens in the body are two interdependent processes, which have aftereffects and depend on the preexisting states of the cell and virus populations. Different scenarios with the same input parameters are necessary to consider for modeling the properties of the states. The efficiency of the immune response is the most important factor, including the time it takes to develop defense responses from three levels of the immune system, which is a complex system of the body. A computational description of infection scenarios was proposed on the basis of a delay differential equation with two values of the time lag. In the new model, transitions between phases of infectious disease depend on the initial virus dose and the delayed immune response to infection. A variation in the dose of the virus and response time can lead to a transition from an acute phase of the disease with overt symptoms to a chronic phase or fatal outcome. Asymptomatic transmission of viral infection was calculated and described in the model as a situation where the virus is rapidly and efficiently suppressed after a short replication phase, while still persisting in the body in minor amounts. An analysis of the model behavior is consistent with the theory that the initial number of virions can affect the quality of the immune response. The reasons that high individual differences are observed in the trajectory of COVID-19 disease and the formation of types of the immune response to coronavirus are still poorly understood. Known trajectories of hepatitis C virus (HCV) infection were used as a basis for model scenarios.

Keywords: COVID-19 variation; Keywords: modeling infectious processes; asymptomatic scenario; delayed regulation; epidemics; hepatitis C chronification model; immune response; initial infectious dose; virus transmission rate.

Fig. 1.

Comparison (by mortality rate) of the development of the Spanish flu epidemics in 1918 between two American cities.

Fig. 2.

Modeling of the threshold scenario of a population outburst by damping (reproduced from [1]).

Fig. 3.

The relative numbers of coronavirus-infected cases by age group in Iceland and the Netherlands (from https://www.covid.is).

Fig. 4.

(a) A plot of new COVID cases in Texas with a smoothed curve obtained by data averaging over 7-day periods (data from The New York Times) and (b) an oscillating COVID epidemic in Iran.

Fig. 5.

(a) A series of decreasing peaks of the epidemics was observed in Michigan; (b) spontaneously decaying oscillations arise in model (5) after a bifurcation because the parameter r increases.

Fig. 6.

Λ-shaped peaks of an oscillating outburst in model (6) without changes in parameters.

Fig. 7.

The dynamics of damage to forests in two Alaskan regions after the start of an oscillating outburst of the spruce beetle Dendroctonus rufipennis (from a Forest Service report [54]) https://www.nps.gov/articles/insectsswan.htm.

Fig. 8.

Activity peaks of dengue fever carriers in southern China (from [55]).

Fig. 9.

The nondissipative trajectory by which the solution of model (8) passes from the state N(t) ≈ K into the mode N(t) → ∞ at τ = 58, K = 15 000, H = 5000, r = 0.00000335, N(0) = 190, and ξ = 2.

Fig. 10.

The equilibrium that the trajectory of model (8) reaches after N(t) ≈ K in the state N(t) → H; N(0) = 1099.

Fig. 11.

The scenario of asymptomatic chronification of infection with max N(t) < K is observed at a lower regulatory lag in model (8); N(0) = 1200.

Fig. 12.

The dynamics of the HCV RNA concentration and generation of various immune response types as acute infection changes to a chronic scenario (according to [65]).