A Deterministic Model for Understanding Nonlinear Viral Dynamics in Oysters

ABSTRACT Contamination of oysters with a variety of viruses is one key pathway to trigger outbreaks of massive oyster mortality as well as human illnesses, including gastroenteritis and hepatitis. Much effort has gone into examining the fate of viruses in contaminated oysters, yet the current state of knowledge of nonlinear virus-oyster interactions is not comprehensive because most studies have focused on a limited number of processes under a narrow range of experimental conditions. A framework is needed for describing the complex nonlinear virus-oyster interactions. Here, we introduce a mathematical model that includes key processes for viral dynamics in oysters, such as oyster filtration, viral replication, the antiviral immune response, apoptosis, autophagy, and selective accumulation. We evaluate the model performance for two groups of viruses, those that replicate in oysters (e.g., ostreid herpesvirus) and those that do not (e.g., norovirus), and show that this model simulates well the viral dynamics in oysters for both groups. The model analytically explains experimental findings and predicts how changes in different physiological processes and environmental conditions nonlinearly affect in-host viral dynamics, for example, that oysters at higher temperatures may be more resistant to infection by ostreid herpesvirus. It also provides new insight into food treatment for controlling outbreaks, for example, that depuration for reducing norovirus levels is more effective in environments where oyster filtration rates are higher. This study provides the foundation of a modeling framework to guide future experiments and numerical modeling for better prediction and management of outbreaks. IMPORTANCE The fate of viruses in contaminated oysters has received a significant amount of attention in the fields of oyster aquaculture, food quality control, and public health. However, intensive studies through laboratory experiments and in situ observations are often conducted under a narrow range of experimental conditions and for a specific purpose in their respective fields. Given the complex interactions of various processes and nonlinear viral responses to changes in physiological and environmental conditions, a theoretical framework fully describing the viral dynamics in oysters is warranted to guide future studies from a top-down design. Here, we developed a process-based, in-host modeling framework that builds a bridge for better communications between different disciplines studying virus-oyster interactions.


S1: Stability analysis of the long-term equilibrium
For Group 2 viruses, it can be easily seen that the equilibrium is stable using the linear stability analysis. Setting ̇= , it is shown that ̇= − ( + ), which is always below zero, i.e., ̇< 0. This suggests that any small perturbation from the equilibrium * , ( − * ), decays with time so that * = + (equation 20) is always stable, and with any initial condition will approach this equilibrium.
For Group 1 viruses, studies on the stability have been conducted for viral dynamics in a closed system. Our model collapses to a closed system if there are no oyster filtration behaviors that exchange viruses between the oyster inner environment and the surrounding water (i.e., = = 0). For a closed system, it has been suggested that whether the equilibrium * is stable depends on the basic reproduction number 0 (e.g., 1-3). If 0 > 1, * is positive and asymptotically stable (i.e., leading to chronic infections); If 0 < 1, there is no positive * and V decreases to zero (i.e., virus is totally removed). When can be neglected, the expression of 0 for equations 1-3 is (4): When the filtration behaviors are included, the system becomes an open system and it is more complicated to analyze the stability of the long-term equilibrium * in equation 5 or 19.
It is expected that if the filter-in virus is much less than the in-host production during the long period after the acute infection phase, the dynamics of the open system may be similar to that of the closed system, and we may neglect the term . In this case, the equilibrium * is close to * ≈ + , as ≫ .
In this case, the stability analysis for the open system with weak exchange resembles that for the closed system, and there exists a 0 to determine whether * is stable. If is further neglected, then 0 is close to that for the closed system shown above. If 0 > 1, V approaches * that is asymptotically stable. If 0 < 1, oyster has the ability to remove all of the in-host viruses, however, due to the existence of the filter-in process, V remains low but cannot become vanished unless = 0.
If the filter-in is the dominant process for increasing the in-host viral concentrations after the acute infection phase, then the dynamics of Group 1 viruses after the acute phase is similar to that of Group 2 viruses and the equilibrium * is close to * ≈ + , as ≪ .
This indicates the equilibrium * for Group 1 viruses in this case is positive and stable.
More studies are needed for evaluating the stability of the long-term equilibrium for Group 1 viruses, especially for the conditions that filter-in and in-host production are both important after the acute phase.
4 Fig S1. The in-host model and its transformed forms in this study. ̂, ̂, and ̂ have the units of cells/oyster, cells/oyster, and virus copies/oyster, respectively. and the conversion factors to convert the units of copies/oyster and cells/oyster to the targeted units, respectively. m is the weight of the total target cells in one oyster and has the units of g/oyster. is the total target cells if the oyster is not infected, and ′ and ′ (unitless) are normalized and , respectively.