The logic of coronavirus infection: revealing the heterogeneity of disease progression and treatment outcomes in COVID patients

doi:10.21203/rs.3.rs-4079008/v1

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The logic of coronavirus infection: revealing the heterogeneity of disease progression and treatment outcomes in COVID patients

https://doi.org/10.21203/rs.3.rs-4079008/v1

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Pneumonia caused by coronavirus infection is a self-limiting disease. Its progression and prognosis are highly heterogeneous among people of different ages, genders, and living with different life styles. Such heterogeneity also exists in treatment outcomes of different patients. Various physiological and pathological factors, such as renewal of pulmonary cell, number of entry receptor and viral replication, have been identified linking to the development of the disease. However, it is still unclear how these factors collectively establish a causal relationship in the course of disease progression. In this study, we built a mechanistic model to explain the dynamics of infection and progression of coronavirus disease. We modeled how the interaction of pulmonary cells determine the dynamics of disease progression by characterizing the temporal dynamics of viral load, infected and health alveolar cells, and dysfuctional alveolar cells. The viral and cellular dynamics captured different stages of clinical manifestations in individual patient during disease progression: the incubation period, mild symptom period, and severe period. We further simulated clinical interference at different stages of disease progression. The results showed that some medical interventions show no improvement either in reducing the recovery rate nor shortening the recovery time. Our theoretical framework may provide a mechanistic explanation at the systems level for the progression and prognosis of coronavirus disease as well as other similar respiratory tract diseases.

Biological sciences/Computational biology and bioinformatics

Biological sciences/Computational biology and bioinformatics/Computational models

Biological sciences/Microbiology/Virology/Sars cov 2

Biological sciences/Microbiology/Virology/Viral vectors

Coronavirus

Infection Heterogeneity

Disease Progression

Mechanistic Model

Clinical Intervention

Coronavirus disease (COVID) is a self-limiting disease, but it sometimes can develop into conditions in which patient suffer sever acute respiratory syndrome (SARS) or acute respiratory distress syndrome (ARDS). One of the key issues remaining elusive is the heterogeneity of disease progression. Firstly, it is still not clear that how the disease suddenly progress from mild to sever condition and what dynamical mechanism contribute to that fast transition. For example, holding for several days of mild symptoms, patients’ health condition can rapidly deteriorate within one or two days. The severe cases need extra mechanical ventilation or even extracorporeal membrane oxygenation ^1–3. This has been associated with the accumulation of cytokines and cytotoxic lymphocytes which cause massive pulmonary damage. In particular the disease severity is attributed to the lymphocytes that is cross-reactive to the coronavirus that cause seasonal flue ⁴. In addition, disease progression showed distinct developmental patterns in people of different age structure⁵, life-style (eg. smoker versus nonsmoker)^6,7 and biological sex ^8–10. For example, in the early stage of the pandemic, the majority of infected children and young adults develop no or mild symptom ^11,12. The clinical manifestations of infections are less severe and recovered shortly without medical intervention. However, infection in elder groups (usually elder than 60) are more likely to develop sever and critical illness. Epidemiological survey across countries fosters worse infection outcomes in males compared to females. Studies show many factors including receptor density, viral load, and accumulation of interferon are related to the development of the disease. For example, the disease progression is associated with low abundance of lymphocytes, typically CD4⁺ and CD8⁺ cells ^13,14. The causative connection between these factors and disease progression is still lacking. Although many physiological and serological indicators are associated with the severity of this disease. On the individual level, however, it is rather difficult to predict whether and when a mild case will transit into sever or critial condition. We believe that it is important and necessary to establish a quantitative framework to explain the heterogeneity of infection outcomes and disease progression of Coronavirus disease.

Accumulating evidence showed that the excessive immune response followed by viral infection is the cause of pathogenesis of COVID, especially in critically ill patients. Although viral load is associated with severity, significantly reduced viral load is observed in critically ill patients or severe phase of disease progression ¹⁵, This trend was also observed in other viral infection, including human influenza A (H5N1) ¹⁶ and SARS-COV ³. For example, type III interferons in lower respiratory tract induces barrier damage and cause susceptibility to lethal bacterial superinfections ¹⁷. In patients at different stage of COVID, cytokine and chemokine profiling in peripheral blood revealed that the pathogenesis are associated with several markers of endothelial injury and thrombosis, such as IL-6 and GM-CSF ¹⁸. High level of cytokines are also presented in severely ill patients of many other viral disease ^3,16,19. Inhibition of cytokines in clinical treatment significantly prevented disease progressing into critical conditions ²⁰. In addition, Some types of T cells many play an important role in deterioration of diseases, in particular, those cross-reactive among many other coronaviruses with high level of cytotoxicity^4,21,22.

Various treatment strategies have been implemented to treat COVID during the pandemic and also resulted in remarkably varied outcomes. For example, treatment with protease inhibitors lopinavir and ritonavir showed no benefit compared to standard care in severely ill patients ²³. Meanwhile, a traditional chinese medicine formula Lianhuaqingwen showed significant antiviral capacity in vitro, improved recovery rate and shortened treatment time of mild cases. However, the formula has no effect in severe cases^24,25. Many other antiviral treatments also revealed similar patterns of treatment both in vivo and in vitro ^26–28. This indicates that the treatment outcome may depended on therapeutic window.Theoretical analysis showed that treatment through inhibiting viral replication should be as early as possible ^29,30, which has been confirmed by subsequent clinical trails ^24,31,32. Virus neutralizing treatment with convalescent plasma showed no benefit in a randomized trail ³³, but a latter trail revealed that early administration with high titter resulted better outcomes ³⁴. The underlining mechanism that resulted in varied outcomes is still unclear.

In order to gain a quantitative understanding coronavirus disease progression, we established a simple theoretical framework to investigate how various factors collectively contributes to the heterogeneity of disease progression and treatment outcomes. In doing so, we firstly analyzed the cellular dynamics of pulmonary cells in the lung (See Fig. 1), then build a mathematical framework to capture the cellular dynamics of health and virus-infected alveoli. Pulmonary alveoli are consists of two types of cells, type I and type II cells. Pulmonary renewal is sustained by the division of type II cells. Resident macrophages scatter on inner surface of alveoli and romove invading pathoge, dead cells and inhibiting inflammation ³⁵. Such dynamics maintains pulmonary homeostasis. When infected by coronavirus, the alveolar cells suffer impaired growth and division. We assume that virus infection will not immediately cause cellular death and apoptosis. Instead, it will deplete heath cells with a measurable rate and render the infected cell into apoptotic cells, whose cytokines releasing will cause severe immune-mediated pathology with presence of cross-reactive cytotoxic T cells. With the mechanistic framework, we then modeled treatment with viral entry inhibitors, viral proliferation inhibitors, cytokines inhibitors and combined treatment. We analyzed how the dosage and treatment window will affect the outcome and recovery variation. With above framework, we hope to establish a dynamic process to explain how the virus and pulmonary cells interact together and determine the outcome of an infection. Moreover, this model can provide some theoretical guidance for the treatment of coronavirus and other virus-associated disease.

Cellular dynamics of alveolar cells in healthy lung

In order to understand the dynamics of pulmonary pathogenesis under coronavirus infection, we firstly investigate the maintenance of alveolar homeostasis in healthy condition. There are mainly two types of alveolar cells that compose alveoli: type I alveolar cell, type II alveolar cell ³⁶. Type II cells proliferate and differentiate into type I cells with a proportion of $ϵ$. Thus, the dynamics of type II and Type I cells, ${C}_{II}$, ${C}_{I}$, can be written as

$$\frac{d{C}_{II}}{dt}=\alpha (1-ϵ){C}_{II}-{\delta }_{x}{C}_{II},$$

$$\frac{{dC}_{I}}{dt}=\alpha ϵ{C}_{II}-{\delta }_{x}{C}_{I}.$$

We assume that all types of cells have the same rate of aging or apoptosis, ${\delta }_{x}$. The resident macrophages on the surface of alveoli are able to self-renew with rate $\beta$. We have the dynamics of resident macrophages, ${M}_{\varphi }$,

$$\frac{d{M}_{\varphi }}{dt}=\beta {M}_{\varphi }-{\delta }_{x}{M}_{\varphi }.$$

These macrophages play important roles in removing invading pathogens and apoptotic cells, and inhibiting inflammation ³⁷(Fig. 1B). Aged or apoptotic cells can be removed by macrophages with rate ${\varphi }_{C}{M}_{\varphi }$. We have the dynamics of apoptotic cells,

$$\frac{d{C}_{x}}{dt}={\delta }_{x}({C}_{II}+{C}_{I}+{M}_{\varphi })-{\varphi }_{C}{M}_{\varphi }{C}_{x}.$$

Notably, if $d{C}_{x}/\text{d}\text{t} <0$, The apoptotic cells can always be removed and the hemeostasis of alveolar cell growth is maintained.

The dynamics of viral infection and disease progression

Due to the wide distribution of angiotensin-converting enzyme 2 (ACE2) in tissues and organs, SARS-COV-2 can enter and infect most of, if not all, the tissues and organs ³⁸. Alveolar cells are early encounters during the course of infections. We divided alveolar cells into three groups when infected with coronavirus: healthy cells, infected cells and dysfunctional cells. With above dynamical model of healthy alveoli, we assume that only non-infected type II cells can divide. Infected cells are still able to maintain the pulmonary structure and function of gas exchange despite losing their ability to proliferate. If the viral load decrease, these infected cells are able to turn into healthy cells. We define dysfunctional cells as as a group of cells which are not able to exchange gases. These cells are accumulated as a results of virus-induced pyroptosis, immunopathogenesis and subsequent fluid infiltration. In particular, the infected alveolar cells can secrete interferon to neighbour cells and enhance their antiviral capacity, which can substantially reduce viral load. In case of uncontrolled viral infection, cytokines release from dead and dysfunctional cells to peripheral blood and attract cytotoxic lymphocytes to the infection site. The resulted massive lung damage substantially impair gas exchange in the lung and cause severe clinical manifestations (see Fig. 1C). Specifically, we decompose the whole process of infection and manifestation as follows:

viral infection: All alveolar cells are infected by the virus with same rate $\sigma V$, with which uninfected cells will be removed. In addition, infected cells can recover depending on the viral load. We have the dynamics of uninfected cells, ${C}_{II}$, ${C}_{I}$ and ${M}_{\varphi }$,

$\frac{d{C}_{II}}{dt}=\alpha (1-ϵ){C}_{II}-\sigma V{C}_{II}+{\lambda }_{1}r(1-f(V\left)\right){C}_{\iota },$	(5)
$\frac{d{C}_{I}}{dt}=\alpha ϵ{C}_{II}-\sigma V{C}_{I}+{\lambda }_{2}r(1-f(V\left)\right){C}_{\iota },$	(6)
$\frac{d{M}_{\varphi }}{dt}=\beta {M}_{\varphi }-\sigma V{M}_{\varphi }+{\lambda }_{3}r(1-f(V\left)\right){C}_{\iota }.$	(7)

Here,we assume that infected cells are not able to proliferate. The apoptotic rate of health cell in uninfected condition, ${\delta }_{x}$, is comparably low by assuming that ${\delta }_{x}\ll \sigma V$, thus can be neglected. If virus inside the cell can be suppressed and eliminated, the infected cells will turn into healthy ones with rate $r(1-f(V\left)\right)$, where $r$ is a rate parameter, and $f\left(V\right)=\frac{{V}^{n}}{{V}^{n}+{\varTheta }_{V}^{n}}.$ V is the viral load and ${{\Theta }}_{\text{V}}$is the half saturated threshold of viral load, $n$ is the Hill coefficient. ${\lambda }_{i}$ is the ratio factor of each type of cell, with $\sum {\lambda }_{i}=1$.

cell death and removal: Coronavirus infection will not immediately lead to cell death, but can lose ability of division, with rate ${\delta }_{x,vir}$ due to viral pathogenesis. In case of cytokines storm, these infected cells, ${C}_{\iota }$, can be directly killed by cytotoxic lymphocytes ${C}_{T}$, if any, due to the presence of viral antigens on the surface. The removal rate denotes ${\varphi }_{T}{C}_{T}$. The dynamics of infected cells are described as

$$\frac{d{C}_{\iota }}{dt}=\sigma V({C}_{II}+{M}_{\varphi }+{C}_{I})-{\delta }_{x,vir}{C}_{\iota }-{\varphi }_{T}{C}_{T}{C}_{\iota }\left(\frac{{I}_{l}^{n}}{{I}_{l}^{n}+{I}_{l,50}^{n}}\right)-r(1-f(V\left)\right){C}_{\iota }.$$

Dead cells,${C}_{x}$, are accumulated from the killed infected cells, and can be removed by uninfected macrophages. The dynamics is

$$\frac{d{C}_{x}}{dt}={\varphi }_{T}{C}_{T}{C}_{\iota }\left(\frac{{I}_{l}^{n}}{{I}_{l}^{n}+{I}_{l,50}^{n}}\right)+{\delta }_{x,vir}{C}_{\iota }-{\varphi }_{C}{M}_{\varphi }{C}_{x}\left(\frac{{I}_{f}^{n}}{{I}_{f}^{n}+{I}_{f,50,C}^{n}}\right).$$

viral replication and immune response: Virus replicates inside the infected cells. We calculate the virus growth rate as average number of viral particles, thus the rate of viron production writes $\gamma {C}_{\iota }$, where ${C}_{\iota }$ is the population size of infected cells, $\gamma$ is the number of viral particles each cell produces. Viral replication can be diminished by cellular regulation with interferon signals³⁹. The viral growth can be written as

$$\frac{dV}{dt}=\gamma {C}_{\iota }-{\varphi }_{V}\left(\frac{{I}_{f}^{n}}{{I}_{f}^{n}+{I}_{f,50,V}^{n}}\right)({C}_{II}+{C}_{I}+{M}_{\varphi })V.$$

Where ${\varphi }_{V}$ is the rate that health cells clear virus. ${I}_{f}$ is the type I cytokines, such as interferon, which is produced by infected cells, and removed with constant rate ${\delta }_{I}$,

$$\frac{d{I}_{f}}{dt}={\rho }_{f}\left(\frac{{C}_{\iota }^{n}}{{C}_{\iota }^{n}+{\varTheta }_{\iota }^{n}}\right)-{\delta }_{I}{I}_{f}.$$

Massive infection and cellular dysfunction cause cytokines storm and self-targeting immune response. These cytotoxic immune cells are mainly from peripheral blood and attracted to the infected site by cytokines. Here we do not consider the loss-of-function of cytotoxic immune cells when infected by virus, and only consider functional cytotoxic immune cells, ${C}_{T}$, that arrive at infected site. The rate equation of type II cytokines can be written as

$$\frac{d{I}_{l}}{dt}={\rho }_{l}\left(\frac{{C}_{x}^{n}}{{C}_{x}^{n}+{\varTheta }_{x}^{n}}\right)-{\delta }_{I}{I}_{l}.$$

${\varTheta }_{i}$ ($i$ = {$\iota$, $x$}) is the half-saturated threshold of dead and dysfunctional cells, and $n$ is the Hill coeficient.

Model analysis and implementation of in silico treatment

In order to specifically study the impact of kinetic factors on the pathogenesis and treatment of coronavirus, we performed detailed numerical simulation and analysis of the model. We define the “in silico clinical manifestation” according to the temporal patterns of cell numbers and viral load. In the course of infection, when the number of infected cells is less than the number of healthy cells, we define the time interval as the asymptomatic phase; when the number of infected cells is greater than the number of healthy cells, we define the time interval as the symptomatic phase; when the number of dead and dysfunctional cells is more than the functional cells, we define the time interval as severe phase. In order to study how the factors collectively determine the infection outcomes, we tested the effects of several parameters including alveolar cell renewal, macrophage phagocytosis and interferon production on the disease progression.

Various strategies showed potency in treating COVID-19, which includes preventing viral entry ⁴⁰, inhibiting viral replication with antiviral drugs ³², directly neutralizing viral particles through monoclone antibodies^41,42 or convalescent plasma^33,43, eliminating inflammatory cytokines in peripheral blood ^41,44. In above dynamical system, we modeled and parameterized several critical steps of disease progression. The most important three of which are viral entry with rate $\sigma$, viral replication with rate $\gamma$ and cytokines production with rate $\rho$. We further in-silico investigated how the clinical treatment could mitigate the disease progression and recovery. We simulated three main treatment strategies: reducing viral load, inhibiting viral entry and inhibiting over-reactive immune response induced by cytokines. For the sake of simplicity and proof-of-principle, we do not consider the pharmacodynamics and pharmacokinetics by mimicking constant concentration and manipulating the rate parameters. For example, changing the parameter $\gamma$ into 0.1$\gamma$ is considered rate of viral replication is reduce to 10% of its maximal rate. To test time of treatment implementation, we change parameter value at different time point in the course of disease progression. To test the effect of these treatment strategies in a cohort of patients with heterogeneous disease progression, we firstly tune above three parameters and generate “in silico disease manifestation” in individual patient according to above definition. In simulating cohort treatment, we firstly created a 70% severity rate in 2000 patients by randomly setting rate of phagocytosis ${\varphi }_{C}$, infection rate $\sigma$ and rate of cytokine production $\rho$ with distributions 10^U(-6,-2), 10^U(-9,-7) and 10^U(-5,-3), respectively. We calculate the average time of asymptomatic phase, symptomatic phase and severe phase of the 2000 patients. We then introduced the treatment strategies with different regimes and their combination at different phases and accessed the recovery rate and cure time. The recovered cases are collected when viral load is less that 10^3 at the end of treatment. All calculations and simulations are performed in R 4.0.4.

A mechanistic model of coronavirus infection and disease progression

In this study, we have built a mechanistic model that mainly captures the dynamics of alveolar cells, viral infection, cellular immune responses mediated by interferons, cytokines and cytotoxic T cells (Fig. 1). Our mechanistic model focuses on the dynamics of these interactions and provides insights into the immune response against coronavirus infection in the lungs. It should be noted that our model is established on the base that there is a physical isolation between pulmonary alveoli and peripheral blood. This physical isolation can be weakened when viral infection cause cellular pyroptosis and release cytokines. This indicate that pulmonary alveoli function independently in terms of immune response for fighting against coronavirus infection, as pulmonary alveoli have resident macrophages that elininate the invading pathogens. In addition, viral infection can be also eliminated by cellular immunity that mediated by interferon produced by neighbour cells (Fig. 1C). When the pulmonary infection can not be controlled by above mechanism, the pyroptotic cells that are caused by infection will release other specific cytokines to attract cytotoxic lymphcytes to the infection site, which might cause severe symptoms.

The dynamics of alveolar cells, viral load and progression of cronavirus disease.

Here, we firstly analyzed the cellular dynamics of alveolar cells in both healthy and infected condition (Fig. 1), then connected the temporal variation of alveolar cells with disease progression and symptom manifestation. Viral infection does not immediately manifest symptoms if the inflammatory response is not initiated. In the case of covid-19, severe symptoms are mostly the consequences of over-reactive immunity that target the patients’ tissue and organ. Stronger cellular response that mediated by interferon can more quickely eradicate virons inside cells. Lymphocytes profiling showed that cross-reactive T cells may be mainly responsible for the sever symptoms. In the simulation, we thus tuned two parameters ${C}_{TO}$ and $\rho$ which represent initial cross-reactive cytotoxic T cells and rate of interferon production. The dynamics revealed in our model defined three different statuses corresponding to real-world infection: (1 no infection, (2 infection with mild symptoms and (3 infection with severe/critical illness (Fig. 2A-C), respectively.

In Fig. 2A, we define this scenario as “no infection” as the viral population is rapidly eliminated and almost no cell is infected and killed. This may be due to fast cellular immune response($\rho$ = 10^-1) and low availability of cross-reactive cytotoxic lymphocytes(${C}_{TO}$ = 1). When infection demonstrated no or mild symptoms (${C}_{TO}$ = 1, $\rho$ = 10^-4.8), the virus propagate and infected cell some of alveolar cells, but did not kill these cells or cause overwhelming inflammatory response, and showing a typical characteristics of self-limiting disease (Fig. 2B). Specifically, we consider the recover process as an intrinsic process, in which issue damage is associated with viral load. When viral load decrease, the lung cells will automatically recover. In clinical cases, the loss of function in lung cells are mainly caused by fluid infiltration, and the function of lung cells can be retained when the infiltrated fluid is absorbed. When infection develops severe symptoms (${C}_{TO}$ = 10^, $\rho$ = 10^-5), viral proliferation and subsequent pathogenesis causes cell death and elicits strong immune response. Alveolar cells was infected by viral particles and loose function due to excessive inflammatory response. In clinical manifestations, this case can be characterize by massive fluid infiltration in the lung. Based on the cellular and viral dynamics, our model can further captures the stages of clinical manifestations in individual patients, e.g. asymptomatic phase, symptomatic phase and recovery phase/severe phase (Fig. 2. D, E). In particular, our model predicts that the transition from symptomatic phase to severe phase is rather swift (Fig. 2. E), which usually lasts 1–2 days in early cases that infected by earlier viral strains ⁵.

Decomposing the complex factors contributing to the infection outcomes

With above definition of infection outcomes, we further investigated how the host, pathogen and other factors collectively determine the outcome of the infection. We identified and scanned the parameters of four main factors that related to the process of infection, namely rate of viral infection, rate of viral clearance, rate of pulmonary cell proliferation, rate of phagocytosis of resident macrophages, and the availability of cross-reactive cytotoxic T cells.

By manipulating two of these parameters each time, we generated the phase diagrams (Fig. 3) for each pair of parameters. In this way, we were able to know how a single parameters as well as parameters pairs decide the trajectory of disease progression. Consequently, we generally find three to four different infection outcomes in different parameter spaces (Fig. 3). The result of simulation showed that the rate of pulmonary cell proliferation does not change the consequence of infection within a large rang of parameter space from 10^− 5 to 1. In addition, our modelling results showed that phagocytosis has little impact on the progression of disease in certain parameter space, as the parameters space (from 10^− 5 to 10^− 2) explored in our simulation did affect the infection outcomes. When the phagocytosis rate is larger than 10^− 2, it results better outcome of infection. It is intuitive that higher infection rate leads to worse disease outcomes. When the infection rate increase from 10^− 9 to 10^− 8, the in-silico “patients” have severer symptoms. Simlarly, increase in virus removal in early stage of infection will drastically increase the chance of recovery without medical interference. As the rate of clearance increase, the infection outcomes change from “infection with recovery” to “asymptomatic infection”.

Besides, we also found that the disease development trajectory of the “patients” was closely related to the availability of cross-reactive lymphocytes. In our model, this is the key parameter that determine whether disease will develop into severe and critical stage. We explored three parameters for cross-reactive lymphocytes (10⁴, 10⁵, 10⁶). Infected “patients” with high initial cross-reactive lymphocytes are more likely to develop severe symptoms and fail to recover from the infection.

Medical intervention: strategy and time window

A variety of treatment strategies have been applied to fight against the virus and its causal pneumonia. Many of these treatment strategies provided plausible solutions for clinical cases. We carried out cohort of 2000 random simulations with different disease development trajectory by tuning parameters of virus infection rates, cell proliferation rates and cross-sensitive T cells. We set the average ratio of patients with critical symptoms is about 30% if the “patients” cohort received no medical intervention. The average recovery time is about 80 days with the standard that no virus is detected in patients. When adopting different treatment plans to intervene and treat patients, we use the simulation results as a reference to evaluate the effects of the treatment plan and treatment time accordingly.

In Fig. 4, we found remarkable differences in recovery rate and in-hospital duration when implemented these different strategies. In general, the simulation results indicated that medical intervention should start as early as possible in the course of infection. We find that earlier intervention results higher treatment success but longer in-hospital duration. Applying single treatment strategy, the medical intervention started at the onset of infection or “Symptomatic phase” by our definition can nearly “cure” half of the infected patients. Moreover, it is worth to point out that applying single treatment strategy is unlikely to cure all the infected patients.Our simulations support the combined treatment as a rule of sum in treating Coronavirus disease. Combined treatment with above three individual methods resulted considerably higher chance of treatment success as well as shorter in-hospital duration. In addition, our simulation also showed that implementation of treatment in late stage of infection still lead to worse outcomes. There is also a “trade-off” when implementing these treatment strategies as early treatment results longer in-hospital duration (Fig. 4).

SARS-CoV-2 infection typically originates in the lungs before disseminating to the peripheral blood and various organs and tissues. The viral load in the lungs is influenced by the infection rate, turnover rate of alveolar cells, and cellular immune response. Our study presents a mechanistic model that incorporates these dynamics to comprehensively understand the progression of coronavirus disease and its treatment outcomes. Our model highlighted several key separate and coherent steps that determine the trajectories and outcomes of the infection. First, our model assumption is based on the physical barrier between pulmonary environment and peripheral blood, which ensures that the lung is able to maintain its self-sustained immune mechanism to eradicate the infection. Due to the fact that the physical barrier can separate the pulmonary immunity and humoral immunity, the signal of infection can be barely transmitted to the peripheral blood at the early stage of infection. When healthy lung is exposed to coronavirus, alveolar cells including resident macrophages will be infected. Viral infection can greatly impair cell division and cause loss of cell function and cell death caused by pulmonary embolism if the infection can not be curbed. Then, unstoppable viral infection and cell death cause a large number of cytokines and chemokines to be produced and released into the peripheral blood, thereby attracting cytotoxic lymphocytes to the infected site. This may result severe symptoms or multiple organ dysfunction with similar mechanism.

By integrating above important assumptions, our model captures many features of disease progression and clinical manifestations. One of the typical ones is the sharp transition from mild symptom to severe or critical symptoms, which is not observed in conventional pneumonia caused by other virus. This sharp transition manifestation is commonly observed in case at the beginning of pandemics in Wuhan ⁵. Our model indicate that this abrupt transition is in fact resulted by the massive release of cytokines from infected cells and infusion by immune factors in the peripheral blood after the collapse of physical barrier between organs and the blood with the presence of cross-reactive cytotoxic T cells ^4,45. Similar modelling assumptions and results has also been implemented and obtained, which showed that IFN response is essential to modulate the transition from asymptomatic to symptomatic in different patients ⁴⁶. For proof of principle, our model however, does not explicitly cover dynamics of all components of immune cells and details of cytokines ⁴⁷. Also, we did not consider the other factors, such as collateral bacterial infection, that contributes to the sharp transition of disease progression ⁴⁸.

Despite the spatial differences in different organs controls the heterogeneity of disease progression, we tuned different parameters and identified several key factors that contribute to the variable disease trajectory. The simulation results have shown that lower infection rate, higher phagocytosis rate and cytokine production all lead to less severe symptoms without treatment. Different combinations of parameters resulted in the phase plain of the symptoms, which is the origin of heterogeneity of disease patterns in individual patients. Infection rate is defined as how likely a viron can enter and infect a cell, which is determined by the receptor angiotensin-converting enzyme 2 (ACE2) for SARS-COV-2. For example, clinical records showed that people in younger age are less likely to develop severe symptoms when infected ^5,49. This may be partially due to the fact that pulmonary epthelia in younger people express less viral receptors ^50–52. Blocking these receptors can achieve much lower infection rate ^40,53,54. In addition, younger people may have higher turn-over rate of the pulmonary cells. Other factors, such as smoking life style, also affect the expression of viral receptors and thus cause different infection outcomes in somkers and non-smokers as smoking increase the expression of ACE2 ⁶. Besides, Our model also captured the dynamics that increase in rate of phagocytosis in lower respiratory tract can result in better infection outcome. As the resident macrophages can substantially reduce neutrophil-driven inflammatory damage ⁵⁵, also inactivate CD4 T cell mediated inflammatory immune response ^35,56,57. In addition, cytokine is one of the most important factors that inhibit the intracellular infection of virus. For example, IFN deficiency could increse host susceptibility to various pathogen infection ³⁵. But, the cytokines may collectively influence the antiviral capacity in a very complex way ⁵⁸. It is notable that cross-reactive cytotoxic lymphocytes are one of the critical factor that cause severe symptoms in corinavirus infection ^59,60. These lymphocytes are the first wave that was attracted by the cytokine storm from lung. Our model showed that pre-existence of the cytotoxic lyphocytes (such as CD8⁺) can significantly increase the chance developing severe symptoms in patients.

Various strategies with different classes of drugs have been developed to treat SARS-COV-2 infection at its different developmental stages. This includes drugs that inhibits viral entry ⁵⁴and replication ^23,24,61. Our model and results of in-silico treatment showed that different treatment can result variable outcomes. Notably, earlier treatment could results better outcomes based on the simulation. During the pandemic, many drugs have been tested in clinical trials, such as lopinavir-ritonavir, as a compound used for competitively inhibiting viral RNA polymerase and blocking viral replication and transmission. Clinical trails showed that drugs only applied in the early stage of infection can result for better treatment outcomes ^23,32. This also happen to the drug hydroxychloroquine that inhibits ACE2 to prevent the coronavirus from entering the cell ^62,63. Although, inhibition of inflammatory cytokines, such as IL-6, can significantly reduce the probability of developing severe symptoms, treatment with Tocilizumab in later stage of the severe cases still can not reverse the disease progression ²⁰. Also, drug combination is widely applied in treating coronavirus disease. Our modelling results showed that combined drug regimes will significantly increase the treatment success when applied at the early stage of disease progression. Similar results have be obtained by theoretical studies with different modelling techniques ³⁰.

Choosing right time window and drug combinations for treating coronavirus disease and other similar disease can significantly improve treatment effectiveness and reduce the total medical costs. However, potential challenges still exist when such optimal strategy is deployed. Our model has suggested the cause of the heterogeneity of disease progression and treatment outcomes, in which the optimal strategy can be explored and implemented. The spatial heterogeneity that caused by the physical isolation between lung and peripheral blood suggests that the disease progression is actually a one-way process, which can be hardly reversed especially when disease develops to late stages. Also, many clinical investigation also have demonstrated that the only way to cure the COVID is to prevent it before it’s getting worse. In addition, this optimal management also rely on proper detection of disease signals. This is particularly important when there is a need for reducing the overall treatment cost in case of an explosive outbreak. Another challenge exists in the current treatment protocols. In many cases, treatment only implemented when the symptom manifests. Our model results are also in line with some prophylactic treatment strategies should be consider for reducing the total costs as the start time of treatment matters the overall outcomes ^64,65.

There are still limitations in our mathematical framework. Despite our model have captured some important features of disease progression and also give practical suggestions for treatment, detailed clinical data and physiological data are still in need for quantitatively predicting the phenomenon and treatment outcomes. For example, the key parameters that the rate of pulmonary cells are killed by viral infection and how fast the apoptosis can attract the immune cells in patients of different ages and physical conditions. In order to capture the dynamics, we although specifically model different kinds of cytokines, we ignore the potential cross-effect of different cytokines, which might be in capable of predicting the treatment effect of different drugs, such as those drugs used for neutralizing cytokines. Moreover, our model did not include the effect of subsequent adaptive immunity. This is because our model only consider the situation where patients get infected with virus for the first time. Future development in models that includes adaptive immunity might gave a more complete picture for this disease. Those includes the effect of vaccination and multiple infections on the development and treatment of disease.

Overall, our mathematical model provides a quantitative framework for disentangling the dynamics of disease progression and its heterogeneity, treatment optimization, especilly enabling us to reveal the pathogenesis of coronavirus disease. Our mathematical framework highlights the importance of deploying the the medication at the early stage of disease development. We hope that this frame work might be also useful for studying disease progression of other viral diseases both in clinics and animal farms.

Table 1

Parameter values used in the simulations.
Parameter/States	Description	Value/Value space
${C}_{II}$	Type II alveolar cells	10¹⁰ (cells)
$\alpha$	Renewal rate of Type II alveolar cells	10^− 3 (cell^− 1.day^− 1)
${C}_{I}$	Type I alveolar cells	10¹¹ (cells)
$ϵ$	Ratio of Type I cells generated by proliferation of Type II cells	0.8(-)
${M}_{\varphi }$	Resident macrophages	10⁹ (cells)
$\beta$	Renewal rate of resident macrophages	10^− 3 (cell^− 1.day^− 1)
${\delta }_{x}$	Rate of health cells turned into dead cells in uninfected condition	10^− 15(cell^− 1.day^− 1)
${C}_{x}$	Dead cells	Set to 0 as initial value
${\varphi }_{C}$	Removal rate of resident macrophages	10^− 3(cell^− 1.day^− 1)
$V$	Viral load	Varied initial values
$\gamma$	Viral proliferation in infected cells	10^− 3(V.cell^− 1.day^− 1)
${C}_{\iota }$	Infected cells	Set to 0 as initial value
$\sigma$	Infection rate of virus	10^− 8(cell^− 1.day^− 1)
${\lambda }_{i}$	Ratio factor of each type of cell	${\lambda }_{1}{=\lambda }_{3} = 0.1$; ${\lambda }_{2}=0.8$ (-)
$r$	Rate of infected cells turned into healthy cells when virus is cleaned	0.1 (-)
${\delta }_{x,vir}$	Death rate of infected alveolar cells	10^− 4(cell^− 1.day^− 1)
${\varphi }_{T}$	Killing rate of cytotoxic lymphocytes	10^− 4(cell^− 1.day^− 1)
${C}_{T}$	Cytotoxic lymphocytes	Varied initial values
${I}_{f}$	Interferon	Varied initial values
${\varphi }_{V}$	Rate that health cells clear virus	10^− 5(cell^− 1.day^− 1)
${\delta }_{A}$	Removal rate of interferon	10^− 3(day^− 1)
$\rho$	Rate of interferon production	10^− 4-10^− 1()
${C}_{T0}$	Cytotoxic immune cells	10¹-10⁴(cells)
$\xi$	Rate of cytotoxic lymphocytes moving to infected site
K₁	Capacity for alveolar cells	10^9
K₂	Capcity for resident macrophages	10^6
n	Hill coefficient	2

Figure legends

Figure 1. The schematic illustration of cellular dynamics of pulmonary alveolus in health condition and under coronavirus infection. A: The anatomic structure of the pulmonary alveolus. The alveoli consist of two types of cells:type I and type II cells. Type I cells maintain the alveolar structure and gas-exchaging function, type II cells are mainly responsible for cell renewal of both types. In addition, there is a tissue-specific macrophage that reside inside the alveolus. B:In health lung, type II alveolar cells proliferate itself and differentiate into type I cells in order to maintain the structure of the lung. The resident macrophage maintains constant division inside alveolus. The resident macrophage can also clear the apoptotic alveolar cells and inhaled microbes, which substantially inhibits local inflammation inside the lung. C: When exposed to coronavirus, alveolar cells including resident macrophage will be infected. Viral infection can substantially impairs the cell division and results enhanced pyropotosis and cell death. Unstoppable infection and cell death release large quantity of cytokines and chemokines into peripheral blood, which elicits proliferation of lymphocytes and attracts them to infection site and sometimes cause lung distruction and sever symptom.

Figure 2. Cell and virus dynamics capture disease progression. Figures A, B, and C show the dynamic of cells and virus in the patient that mark the heterogeneity of infection outcomes: no infection, infection with mild symptoms and infection with severe symptoms, respectively. When the patient enters the mild stage, the number of cells with normal functions in the body decrease significantly, and the infected cells and viral load increase; in the severe stage, the infected cells and viral load exceed normal cells, and a large number of cells die or loosing function of gas exchanging. Figures D and E show the changes in healthy cells, infected cells, and cell populations that have lost function in different periods. When the number of infected cells can be gradually reduced, and the number of pyroptotic cells remains stable and does not rise, the patient enters the recovery period instead of developing into a severe illness.

Figure 3. The phase diagram of infection outcomes with range of parameters of rate of viral infection, rate of viral clearance, rate of pulmonary cell proliferation and rate of phagocytosis of resident macrophages.

Figure 4. The impact of different medical interventions along the time of disease progression. The upper panel represents the proportion of severe cases and the lower panel represents the rate of recovery before and after treatment. The horizontal black lines are there average values of a cohort of in-silico “patient” before treatment. The dark blue dots represent the time points when the intervention is implemented. The treatment window spans from symptomatic phase to severe or critical phase.

Author Contribution

G.Y. and H.H. wrote the main manuscript and prepared figures.

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No competing interests reported.

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The logic of coronavirus infection: revealing the heterogeneity of disease progression and treatment outcomes in COVID patients

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Abstract

Figures

Introduction

Methods

Cellular dynamics of alveolar cells in healthy lung

The dynamics of viral infection and disease progression

Model analysis and implementation of in silico treatment

Results

A mechanistic model of coronavirus infection and disease progression

Decomposing the complex factors contributing to the infection outcomes

Medical intervention: strategy and time window

Discussion

Declarations

Author Contribution

References

Additional Declarations

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\(\frac{d{C}_{II}}{dt}=\alpha (1-ϵ){C}_{II}-\sigma V{C}_{II}+{\lambda }_{1}r(1-f(V\left)\right){C}_{\iota },\)	(5)
\(\frac{d{C}_{I}}{dt}=\alpha ϵ{C}_{II}-\sigma V{C}_{I}+{\lambda }_{2}r(1-f(V\left)\right){C}_{\iota },\)	(6)
\(\frac{d{M}_{\varphi }}{dt}=\beta {M}_{\varphi }-\sigma V{M}_{\varphi }+{\lambda }_{3}r(1-f(V\left)\right){C}_{\iota }.\)	(7)

Parameter/States	Description	Value/Value space
\({C}_{II}\)	Type II alveolar cells	10¹⁰ (cells)
\(\alpha\)	Renewal rate of Type II alveolar cells	10^− 3 (cell^− 1.day^− 1)
\({C}_{I}\)	Type I alveolar cells	10¹¹ (cells)
\(ϵ\)	Ratio of Type I cells generated by proliferation of Type II cells	0.8(-)
\({M}_{\varphi }\)	Resident macrophages	10⁹ (cells)
\(\beta\)	Renewal rate of resident macrophages	10^− 3 (cell^− 1.day^− 1)
\({\delta }_{x}\)	Rate of health cells turned into dead cells in uninfected condition	10^− 15(cell^− 1.day^− 1)
\({C}_{x}\)	Dead cells	Set to 0 as initial value
\({\varphi }_{C}\)	Removal rate of resident macrophages	10^− 3(cell^− 1.day^− 1)
\(V\)	Viral load	Varied initial values
\(\gamma\)	Viral proliferation in infected cells	10^− 3(V.cell^− 1.day^− 1)
\({C}_{\iota }\)	Infected cells	Set to 0 as initial value
\(\sigma\)	Infection rate of virus	10^− 8(cell^− 1.day^− 1)
\({\lambda }_{i}\)	Ratio factor of each type of cell	\({\lambda }_{1}{=\lambda }_{3} = 0.1\); \({\lambda }_{2}=0.8\) (-)
\(r\)	Rate of infected cells turned into healthy cells when virus is cleaned	0.1 (-)
\({\delta }_{x,vir}\)	Death rate of infected alveolar cells	10^− 4(cell^− 1.day^− 1)
\({\varphi }_{T}\)	Killing rate of cytotoxic lymphocytes	10^− 4(cell^− 1.day^− 1)
\({C}_{T}\)	Cytotoxic lymphocytes	Varied initial values
\({I}_{f}\)	Interferon	Varied initial values
\({\varphi }_{V}\)	Rate that health cells clear virus	10^− 5(cell^− 1.day^− 1)
\({\delta }_{A}\)	Removal rate of interferon	10^− 3(day^− 1)
\(\rho\)	Rate of interferon production	10^− 4-10^− 1()
\({C}_{T0}\)	Cytotoxic immune cells	10¹-10⁴(cells)
\(\xi\)	Rate of cytotoxic lymphocytes moving to infected site
K₁	Capacity for alveolar cells	10^9
K₂	Capcity for resident macrophages	10^6
n	Hill coefficient	2