key: cord-335467-0b0m8v5r
authors: Saha, Asit; Saha, Barsha
title: Novel coronavirus SARS‐CoV‐2 (Covid‐19) dynamics inside the human body
date: 2020-07-19
journal: Rev Med Virol
DOI: 10.1002/rmv.2140
sha: 
doc_id: 335467
cord_uid: 0b0m8v5r

A knowledge‐based cybernetic framework model representing the dynamics of SARS‐CoV‐2 inside the human body has been studied analytically and in silico to explore the pathophysiologic regulations. The following modeling methodology was developed as a platform to introduce a predictive tool supporting a therapeutic approach to Covid‐19 disease. A time‐dependent nonlinear system of ordinary differential equations model was constructed involving type‐I cells, type‐II cells, SARS‐CoV‐2 virus, inflammatory mediators, interleukins along with host pulmonary gas exchange rate, thermostat control, and mean pressure difference. This formalism introduced about 17 unknown parameters. Estimating these unknown parameters requires a mathematical association with the in vivo sparse data and the dynamic sensitivities of the model. The cybernetic model can simulate a dynamic response to the reduced pulmonary alveolar gas exchange rate, thermostat control, and mean pressure difference under a very critical condition based on equilibrium (steady state) values of the inflammatory mediators and system parameters. In silico analysis of the current cybernetical approach with system dynamical modeling can provide an intellectual framework to help experimentalists identify more active therapeutic approaches.

At present, the biggest challenge to the world is the prevention of SARS-CoV-2 spreading by developing an appropriate vaccine. Scientists worldwide are working hard to develop a proper vaccine to counter this deadly virus and human trials are already underway in the United Kingdom, China, Australia, United States, Italy, and Germany.

Chen et al 5 Remdesivir has recently been licensed, but there is currently no other proven effective treatment available for Covid-19 patients, including the immunopathogical components. 8 Guan et al reported that the median incubation period for Covid-19 is 4 days with interquartile range, 2 to 7. The serial interval between two consecutive transmission generation pairs is between 3 days and 8 days. [9] [10] [11] [12] Within 2 days of symptom starts, the SARS-CoV-2 viral load reaches a peak. Severe Covid-19 patients develop acute respiratory distress syndrome (ARDS) on an average of around 8 to 9 days after symptoms when the gas exchange rate inside the respiratory system malfunctions. 13, 14 In the present study, we aim to understand the chain of events after the SARS-CoV-2 virus invaded the human body, creating chaos in the respiratory system, thermostat control, and multiple organ failure systematic networks using the knowledge-based cybernetic model. The benefit of using the cybernetic model is mainly its ability to analyze the capacities and limits of systems components. Virus and its host are within the same system.

In the next section, schematic diagrams (Figures 2-5) show the network of events that leads to cytokine storm, ARDS, loss of thermostat control, and multiple organ failure.

The model analysis then suggests to experimentalists possible measures to control these parameters for a better therapeutic outcome.

In this section, we will discuss the functional pathophysiological changes associated with the human body system as a result of SARS- by budding out of the type-II cell. In this process, the virus replicates itself significantly in numbers, ultimately destroying the host cell. As a result, the host cell produces and releases inflammatory mediators that trigger alveolar macrophages to release cytokines including IL-1, IL-6, and TFN-α. [16] [17] [18] [19] Figure 3 shows that the released cytokines IL-1, IL-6, and TFN-α travel toward the capillary around the alveolus and destroy the endothelial layer inside it that promotes vessel dilation and increases the capillary permeability. As a result, fluid leaks out into the interstitial space (space between capillaries and the cells). This alveolar edema reduces the production of surfactants and increases surface tension, leading to alveolar collapse and impaired gas exchange mechanism.

Patients infected with the virus experience a severe symptom of hypoxemia. Many alveoli become filled up with debris that includes interstitial fluid, macrophages, different proteins, damaged type-I, type-II cells, and some neutrophils resulting in consolidation of the alveolus which alters gas exchange and work of breathing (WOB). 20 F I G U R E 1 Disease spread by single infected human F I G U R E 2 Explains the ways how the virus invades the human respiratory system and uses the existing mechanism of alveolus to replicate its +SS RNA and virus proliferation, inflammatory materials, and cytokines productions 15 F I G U R E 3 Explains the collapsing process of a single alveolus In Figure 4 , the excess cytokines (IL-1, IL-6, and TNF-α) travel to the hypothalamus through the blood vessel and direct it to reset the body temperature to a higher level (fever).

The consolidation and accumulation of alveolar debris inside alveoli produces mucus that stimulates coughing. Covid-19 patients with hypoxemia and increased WOB decrease the partial pressure of oxygen (O 2 ) so stimulating the formation of chemoreceptors, which then trigger the sympathetic nervous system (SNS) to increase heart rate (HR) and increase respiratory rate (RR). 

Based on the facts discussed above, we develop a simple schematic diagram ( Figure 6 ) with accompanying mathematical model using a very simplistic cybernetic approach. It shows that the single RNA strand of the SARS-CoV-2 virus uses type-II cells of the alveolus for its replication. At the same time, it uses existing cellular mechanisms inside the host cell to proliferate in significant numbers. The newly F I G U R E 4 Cytokines instigate hypothalamus F I G U R E 5 Demonstrates the sequence of activities leading to multiple organ failure due to severe inflammation of lungs through systematic inflammatory response syndrome (SIRS), and blood volume decreased formed virus then escapes the host cell and targets another type-II cell in different or same alveolus. During this process, the type-II cell also produces inflammatory mediators that stimulate the macrophage inside the alveolus. As a result, different types of interleukins forms, such as IL-1, IL-6, and TNF-α.

Interleukins then dilate the endothelial layers of the adjacent blood vessel and increase permeability, triggering a cytokine storm and damaging both type-I and type-II cells eventually destroying the entire alveolus. Due to the loss of type-I and type-II cells, the gas exchange mechanism inside the respiratory system becomes dysfunctional. It affects the central nervous system through the hypothalamus because of its anterior connection to the blood vessel, which is full of excessive interleukins. As a result, the hypothalamus sets up a new thermostat control temperature. Due to cytokine storm as well as massive, excessive interleukin production, there could be enormous damage to the blood vessels of the entire body (dilation) that creates a pressure difference.

The proposed mathematical model based on the above schematic network diagram is as follows:

where t stands for time in days, T 1 , T 2 , 

The next events in the sequence are the dynamics of interleukins and type-I cells. The solution of Equations (5) and (2) is

where I 0 N , T 0 1 = baseline initial number of interleukins and type-I cells (steady state) within an alveolus in normal situation before any SARS-CoV-2 virus enters.

Using Equation (10) in Equations (6) to (8), we have the following system:

From Equation (10) we recognize cytokines that can trigger the chaos under certain conditions. We define the term D 0 = β 5 α5 as the pivotal element that can stabilize or destabilize the system under certain conditions.

Cytokine concentrations will be decreasing in time. Eventually, body will not experience further complications even if person is infected with the virus.

As time increases, cytokine concentrations will be heading toward their initial values, the state where there is no infection.

This is the chaotic condition; cytokine storm is inevitable if this condition is fulfilled.

The time-dependent solution to Equations (12) to (14) is as follows:

T H = C TH e −β 7 t + 1

where ₂F1 a, b; c;x ð Þ= P / n = 0 a ð Þ n b ð Þ n c ð Þ n × x n n! and C GE , C TH and C PD are the integrating constants. Apoptosis rate parameters for type-I and type-II cells are calculated in the same way as above from experimental work on alveolar epithelial type II cells on animal models. 21 The experimental work of Kim et al 24 on male and female ferret models infected with the coronavirus collected from Covid-19 confirmed patient in Korea helps us to estimate the growth rate of SARS-CoV-2.

The estimation technique is the same as described above. The inflammatory mediator growth rate has been estimated from the experimental results on chronic intermittent hypoxia of male mice over 4 weeks. 25 Interleukin production rate was estimated based on the experimental results of Channappanavar and Perlman. 26 The gas exchange rate was estimated based on the human lung's consumption of oxygen (O 2 ). In an inactive state, a normal human can consume about 5 to 6 mL per minute at 28 C esophagus temperature. 27 The thermostat control threshold parameter is assumed to be between 37 C < Ω < 41 C 24 . The pressure difference constant is the mean area pressure represents the average pressure (force) that drives blood into the vessels. 28

It is equivalent to 

A mathematical framework has been presented, which characterizes the biokinetic mechanism of SARS-CoV-2 virus and alveoli and the surrounding environment that includes blood vessels around it and F I G U R E 7 Existence of pseudo-steady-state for type-II cells, SARS-CoV-2 virus, and inflammatory mediators for different parameter values within the allowable domains as specified in Table 1 T A B L E 1 Parametric input values of the presented cybernetic model as derived from in vivo sparse data

Growth rate of type-I cells 0.2682 < α 1 < 0.6824 Per day Reference [23] Growth rate of type-II cells 0.0545 < α 2 < 0.3466 Per day Reference [22] Apoptosis rate of type-I cell 0.4274 < μ 1 < 0.5680 Per day Reference [21] Apoptosis rate of type-II cells 0.4274 < μ 2 < 0.5680 Per day Reference [21] Growth rate CoV 2 0.0908 < α 3 < 2.5175 Per day Reference [24] Production rate of inflammatory mediators 0.0100 < α 4 < 0.0145 Per day Reference [25] Production rate of interleukins 1.3271 < α 5 < 2.1060 Per day Reference [26] Gas exchange rate Ψ 7.2 < Ψ < 8.64 L/day Reference [27] Thermostat threshold Ω 37 < Ω < 41 C Reference [24] Mean area pressure P 93.33 mm Hg Reference [28] other organs in the body. The overall impact of the virus influences the fundamental regulation of the host immune system, which leads to a cytokine storm under certain situations, heading toward an instability of the entire body by altering homeostasis drastically, which is chaotic.

The intermediate steady state of type-II cells, SARS-CoV-2, and inflammatory mediators, known as the pseudo-steady-state, play a crucial role in these dynamics.

The pseudo-steady-state Figure 7 shows the pseudo-steady-state of type-II cells, SARS-CoV-2, and inflammatory mediators depending upon the different parameter values within its specified domains, as mentioned in Table 1 . It indicates that as the viral load in the steady-state increase, the type-II cell steady-state values decrease, and inflammatory mediators steady values increase.

In Figure 8 , (Figure 12 ) because the body's temperature is continuously increasing to its highest levels. It means the production of interleukins is so high that it starts damaging the blood vessels around the body, which influences the hypothalamus to reset the temperature to a new high. Now we want to see the pressure difference for If 0 M < D 0 and If 0 M > D 0 . Figure 13 shows if If 0 M < D 0 , an increase of mean area pressure for an initial few days, when it comes down to its normal within 14 days. It means that there is minimal damage to the blood circulatory system by interleukins. The body remains into its normal state with 14 days, even if a virus infects it. Figure 14 shows a continuous fall of mean area pressure in the blood circulatory system when If 0 M > D 0 within 14 days. It means that there is very severe damage to the blood circulatory system by interleukins due to cytokine storm that may incur further damage to other organs inside the body.

The enormous complexity of viral dynamics and burgeoning volumes of laboratory data pose a real challenge to researchers in biology and healthcare to assimilate it as real knowledge to tackle the infection. 29 F I G U R E 1 0 Gas exchange rate when If 0 M > D 0 . It is showing that the exchange rate is continuously going down much below the expected level of the body's gas exchange rate F I G U R E 1 1 Body temperature when If 0 M < D 0 . Body temperature is going up slightly in the beginning, and then it is coming down to normal within 14 days 

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The authors acknowledge the technical assistance of Professor S. N.Sarbadhikari, Expert member, Roster for Digital Health, World Health Organization (WHO) and Professor Paul Griffiths, Professor of Virology, University College London for reviewing and editing.

The authors have no competing interest.

https://orcid.org/0000-0003-3139-1959