Editor's Note: This article originally appeared in SIAM News on February 11, 2022 (https://sinews.siam.org/Details-Page/a-dynamical-systems-view-of-the-outcomes-of-viral-infections).

The COVID-19 pandemic has revealed the remarkable heterogeneity in the outcomes that stem from a viral infection. While some individuals can clear SARS-CoV-2 while remaining completely asymptomatic, others require hospitalization with intensive care and may even succumb to the virus [6]. Such a wide variety of outcomes was recognized well before the pandemic as a hallmark of viral infections, although the extent of heterogeneity may differ with each virus. For instance, approximately 25 percent of those with hepatitis C spontaneously clear the infection. For the remaining patients, the infection turns chronic; if left untreated, it can inflict severe liver injury and cause liver cancer [7]. As another example, adults nearly always clear hepatitis B but children do not [5]. And regardless of patient, HIV-1 infection invariably turns chronic [9]. Therefore, scientists have long wondered what factors determine the outcomes of viral infections. A deeper understanding of these causes could inspire novel approaches for intervention.

The most effective studies of viral infection outcomes have traditionally employed animal models; the infection of mice with the lymphocytic choriomeningitis virus is an especially popular technique [8]. Researchers classify the outcomes into three broad categories: (i) Clearance following an acute infection, (ii) long-term persistence, and (iii) mortality due to immunopathology (i.e., tissue damage that results from an over-aggressive immune response). Several factors are known to influence these outcomes, namely the viral genetic make-up; the amount of virus that causes the infection; and the strengths of several components of the host immune response, including CD8 T cells, innate immune responses, natural killer cells, and so on. The classical view thus asserts that viral infection outcomes are the consequences of a plethora of viral and host factors, which makes them difficult to comprehend and predict [8]. Indeed, scientists do not yet fully understand what causes the different outcomes of SARS-CoV-2 infection.

Figure 1. The dynamical motif that underlies the outcomes of infection. Figure courtesy of Narendra Dixit.

In a recent study, we developed an alternative, contrasting hypothesis to the classical view [1]. By analyzing the vast body of available experimental evidence, we reasoned that viral infection outcomes are determined by the essential interactions of just two entities: virus-infected cells and CD8 T cells. Furthermore, the remaining factors indirectly exert their influence on the outcomes by modulating the essential interactions. Virus-infected cells multiply by producing progeny viral particles that can infect uninfected target cells. When these cells encounter CD8 T cells, the latter become activated and kill the infected cells. This exchange controls the multiplication of infected cells. Yet when the infected cell pool is large and interacts with CD8 T cells over sustained durations, the CD8 T cells suffer exhaustion that limits their ability to kill infected cells. These interactions collectively form a dynamical motif (see Figure 1). A minimal set of equations describes the dynamics of the infection, which is governed by this motif [1]:

\[(\textrm{Infected  cells}) \;\;\;\; \frac{dI}{dt}=k_1I\Bigg(1-\frac{I}{I_\textrm{max}}\Bigg)-k_2IE\tag1\]

\[(\textrm{CD8T  cells}) \;\;\;\; \frac{dE}{dt}=\frac{k_3IE}{k_p+I}-\frac{k_4IE}{k_e+I}.\tag2\]

Here, infected cells \(I\) proliferate with first-order rate constant \(k_1\), which is limited by the logistic term with carrying capacity \(I_\textrm{max}\). The carrying capacity represents the maximum that the population of infected cells can reach, based on the infection of the entire pool of target cells or other similar limitations. Infected cells are killed by CD8 T cells \(E\)—also called effector cells—with a second-order rate constant \(k_2\). The killing is modelled using a mass-action type interaction, assuming that the cell populations are well mixed. The CD8 T cells become activated upon contact with infected cells, with a second-order rate constant \(k_3\) and a half-maximal saturation constant \(k_p\). The saturating function represents other limitations to such activation in the presence of abundant infected cells. In the same way, contact with infected cells can trigger the exhaustion of CD8 T cells with rate constant \(k_4\) and saturation constant \(k_e\). We choose the parameter values so that activation of CD8 T cells dominates exhaustion when the infected cell population is small, whereas the opposite happens when the CD8 T cell population is large.

The long-term outcomes of infection represent the stationary states of this dynamical system. By solving for these stationary states, we realized that the system is bistable (Figure 2). The first stable steady state \((I=0\) and \(E\ge0)\) represents viral clearance, and the second stable steady state \((I=I_\textrm{max}\) and \(E=0)\) represents persistence. A separatrix defines the basins of attraction of these two stable steady states. For a given viral inoculum size or initial pool of infected cells, clearance occurs when the initial effector pool is larger than a certain threshold (which is determined by the separatrix). Persistence results from smaller effector pool sizes.

Figure 2. Phase portrait associated with the dynamical system that equations \((1)\) and \((2)\) describe. The solid red circles are stable states, the open red circle is a saddle, and the red line is the separatrix. Curves with arrows indicate trajectories that lead to the different outcomes of viral infection. Figure courtesy of [1].

Interestingly, immunopathology arises in this dynamical system as an outcome that is driven by the third, unstable stationary state: a saddle point that lies on the separatrix. When the initial conditions are away from the separatrix, trajectories rapidly converge to one of the two stable steady states. When conditions are close to the separatrix, the trajectories exhibit large excursions along the separatrix (due to the saddle) until they approach the saddle and veer off to a stable state. Both \(E\) and \(I\) attain large values during this process, which can cause large-scale tissue damage due to the effector pool’s mass-action-based killing of infected cells.

The dynamical motif thus contains the necessary properties of viral infection outcomes. We show that these properties are representative of several experimental observations and offer a synthesis of otherwise seemingly confounding datasets [1]. We also demonstrate how the effects of the other factors that influence outcomes could serve as embellishments of the aforementioned motif [1]. 

We have since shown that one can use this alternative hypothesis to describe the outcomes of several key infections and treatments in humans. With hepatitis C, for example, a curious observation in recent years is the spontaneous clearance of infection well after treatment was stopped, despite detectable virus amounts at the end of treatment. Based on our motif, we argue that the treatment caused a decline of infected cells that in turn reversed CD8 T cell exhaustion and drove the infection to the basin of attraction of the clearance state [2]. A similar argument may underlie the long-term control of Simian immunodeficiency virus in macaques (which mimics HIV infection in humans) after early, short-term exposure to neutralizing antibodies — although the dynamical system is far more complex [4]. Finally, the above motif—when embellished by the innate immune response—also appears to explain the heterogeneous outcomes of COVID-19 [3].

In summary, our new conceptual view of the outcomes of viral infections can synthesize diverse observations across viruses and hosts, thereby offering a framework that could aid the rational optimization of interventions.

Narendra Dixit presented this research during a minisymposium at the 2021 SIAM Conference on Applications of Dynamical Systems, which took place virtually last year.

References
[1] Baral, S., Antia, R., & Dixit, N.M. (2019). A dynamical motif comprising the interactions between antigens and CD8 T cells may underlie the outcomes of viral infections. Proc. Natl. Acad. Sci. USA, 116(35), 17393-17398.
[2] Baral, S., Roy, R., & Dixit, N.M. (2018). Modeling how reversal of immune exhaustion elicits cure of chronic hepatitis C after the end of treatment with direct-acting antiviral agents. Immunol. Cell Biol., 96(9), 969-980.
[3] Chatterjee, B., Sandhu, H.S., & Dixit, N.M. (2021). The relative strength and timing of innate immune and CD8 T-cell responses underlie the heterogeneous outcomes of SARS-CoV-2 infection. Preprint, medRxiv.
[4] Desikan, R., Raja, R., & Dixit, N.M. (2020). Early exposure to broadly neutralizing antibodies may trigger a dynamical switch from progressive disease to lasting control of SHIV infection. PLoS Comput. Biol., 16(8), e1008064.
[5] Ganem, D., & Prince, A.M. (2004). Hepatitis B virus infection – natural history and clinical consequences. N. Engl. J. Med., 350(11), 1118-1129.
[6] Guan, W., Ni, Z., Hu, Y., Liang, W., Ou, C., He, J., … Zhong, N. (2020). Clinical characteristics of coronavirus disease 2019 in China. N. Engl. J. Med., 382(18), 1708-1720.
[7] Hajarizadeh, B., Grebely, J., & Dore, G.J. (2013). Epidemiology and natural history of HCV infection. Nat. Rev. Gastroenterol. Hepatol., 10(9), 553-562.
[8] Oldstone, M.B.A., Ware, B.C., Horton, L.E., Welch, M.J., Aiolfi, R., Zarpellon, A., … Sullivan, B.M. (2018). Lymphocytic choriomeningitis virus Clone 13 infection causes either persistence or acute death dependent on IFN-1, cytotoxic T lymphocytes (CTLs), and host genetics. Proc. Natl. Acad. Sci. USA, 115(33), E7814-E7823.
[9] Sabin, C.A., & Lundgren, J.D. (2013). The natural history of HIV infection. Curr. Opin. HIV AIDS, 8(4), 311-317.

Narendra M. Dixit is chair of the Centre for BioSystems Science and Engineering and a professor of chemical engineering at the Indian Institute of Science. His research interests are in viral and bacterial infection dynamics, theoretical immunology, drug pharmacokinetics and optimization, and microbial community ecology. He was previously a senior fellow of the DBT/Wellcome Trust India Alliance.