Introduction

Despite having over 2000 years of experience with the influenza virus, humanity has yet to fully elucidate its complex nature [26]. The influenza virus infects epithelial cells in the respiratory tract. Infected cells produce new viruses, which infect more epithelial cells. Infected epithelial cells die, and the immune system clears virus (Fig. 1). Outbreaks of seasonal influenza continue to cause significant morbidity and mortality, especially in high-risk groups (children, the elderly, pregnant women, and immunocompromised patients).

Fig. 1
figure 1

A simple schematic describing influenza viral biology. A pool of target respiratory epithelial cells (T) are infected by free virus (V) at the rate constant β. Infected cells (I) shed viruses at a production rate (p). Free virus is cleared at the rate (c). Infected cells produce cytokines which cause disease symptoms; infected cells are cleared at the rate δ. Antiviral drugs exert their effects by inhibiting viral production or target cell infection

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Mitigating the public health and economic impact of influenza outbreaks is complicated by several factors. First, relatively few antiviral drugs have been approved. The licensed antiviral drugs include M2 blockers (adamantanes: amantadine and rimantadine) and neuramindase inhibitors (NI). The World Health Organization recommends oseltamivir or zanamivir―both neuraminidase inhibitors―to treat influenza (WHO guidelines for pharmacological management of pandemic influenza A(H1N1) 2009 and other influenza viruses, [42]). Second, reports from the last several years have shown increasing drug resistance to both adamantanes and neuraminidase inhibitors [40, 41]. Finally, while vaccination is an important strategy against influenza outbreaks, it is an imperfect solution. While the elderly are some of the most susceptible to influenza, vaccination in this subpopulation offers limited protection due to immunosenescence [32]. Moreover, influenza viruses acquire mutations in the hemagglutinin (HA) glycoprotein antibody sites, thus causing a high risk of vaccine mismatches [10, 44].

Clearly, the pharmacological armamentarium needs to be bolstered with new treatments. To achieve success, influenza drug developers must overcome multiple challenges. The currently approved drugs block specific viral proteins: M2 and neuraminidase. Developing new drugs requires the identification of new drug targets within the influenza viral life cycle [20]. Furthermore, the key patient populations most susceptible to influenza-related morbidity and mortality are challenging to study in clinical trials due to ethical and logistical constraints.

There is a clear need for tools to help minimize the uncertainty in making critical decisions in antiviral drug programs. One solution that has been widely used in many industries, including aerospace, finance, and meteorology, is mathematical modeling and simulation. Embraced by regulators, industry, and academia, modeling and simulation can also help develop safer and more effective antiviral drugs.

While discussing all modeling types that are used in drug development is beyond the scope of this review, three types of models stand out for their ability to advance antiviral drug development: pharmacokinetic/pharmacodynamic (PK/PD) modeling, viral kinetic (VK) models, and pharmacoeconomic/pharmacoepidemiologic models. Pharmacokinetics is “what the body does to a drug” whereas pharmacodynamics is “what the drug does to the body.” PK/PD modeling can help determine optimal dosing strategies for antivirals [6, 7] and evaluate the efficacy of antiviral drug combinations [22].

In immunocompetent adults, the influenza virus’s rapid replication generally results in a short duration infection time course. After infection, the virus grows exponentially, peaks at 2 to 3 days post-infection, and declines exponentially. Viral kinetic models use mathematical equations to describe the changes in viral load with time in an infected patient. [1] Viral kinetic models can reveal important parameters to understanding viral biology including the cell infection rate, viral production rate, and viral clearance rate. Viral kinetic models can be combined with PK/PD models to explore how exposure to antiviral drugs can attenuate influenza symptoms and reduce viral load. [9]

When a flu outbreak occurs, public health officials must determine the best way to mitigate its impact and at what cost. Pharmacoeconomic models have been used to assess the costs of various public health treatment strategies as well as quantify the impact of these interventions on patients’ quality of life. Likewise, pharmacoepidemiologic models have been employed to quantify the changes in susceptible, exposed, infected, and recovered patients during an outbreak as well as the transmissibility of an infectious disease in a given population.

Historically, modeling of infectious diseases has been conducted in discrete silos wherein viral biology, antiviral pharmacology, epidemiology, and health economic impacts were not linked. However, in recent years, an integrated, interdisciplinary approach has emerged that links antiviral PK/PD, epidemiology, and health economics. [23] Uniting these disparate modeling approaches can help shed light on the cost-utility of antiviral therapy under various influenza epidemic scenarios.

Review of VK Models for Influenza

Since the mid-1970s, viral kinetics models have been developed to study the pathogenesis of influenza infections. The first such model―built by Larson et al. [27] ―describes the influenza virus concentration dynamics in mice infected with a strain of H3N2 under various experimental conditions of challenge virus dose, site of deposition, and respiratory tissue. The model consisted of seven linear compartments with five constant rate parameters. Later, Bocharov and Romanyukha proposed a mathematical model to describe the joint reaction of the interferon and immune systems in influenza A virus (IAV) infection in humans, using a delay-differential system with about 60 parameters [4].

It was not until 2006 that the first model of influenza A viral infection dynamics in humans came out [1], starting from a “target cell-limited model” approach that had been developed to understand other types of viruses, such as human immunodeficiency virus (HIV), hepatitis C virus (HCV), hepatitis B virus (HBV), and cytomegalovirus (CMV) [38] with additional components that captured specific characteristics for influenza A virus infection. The series of models was fitted and compared to observed viral titer data from six individual subjects. In the simplest model, a system with three compartments described by three differential equations was proposed. The three compartments represented dynamics of uninfected target cells, productively infected cells, and infectious viral titer, respectively. Next, to account for the observed delay in the production of free virus, two separate populations of infected cells were defined. Lastly, the model included the interferon (IFN)-1 response on the kinetics of viral infection and its inhibition on viral replication within the infected cell, an innate immune response of the host. With the interferon component, the model successfully reproduced the bi-modal viral titer peaks observed for about 50% of patients.

In another work, Handel et al. [19] used a version of the target cell model as well as an immune response model to study how use of NIs might lead to the emergence and spread of NI resistance in the population. The same group also developed a model to capture the dynamic interactions between CD8+ T cells and antigen presentation [18]. Almost at the same time, Hancioglu et al. [17] adopted a similar approach by Bocharov and Romanyukha [4] to unravel the control of the infection by both the innate and adaptive immunity.

It was not until the work by Canini and Carrat [7] that more focus was put into connecting influenza A viral kinetics and respiratory symptom dynamics. Their work introduced natural killer (NK) cells, another component of the innate immune responses, and cytokines into the original viral kinetics model by Baccam et al. [1] and further linked systemic symptoms to the concentrations of cytokines. Using a population approach, the proposed model accounted for intra- and inter-individual variability of relevant parameters based on both viral kinetics and symptom dynamics data from 44 healthy volunteers experimentally challenged with influenza A/H1N1 virus.

With the emergence of new drug candidates, modeling efforts have been made to accommodate distinct mechanisms of actions (MoAs) and/or to integrate data from multiple clinical trials from different anti-influenza A virus agents. For example, Kamal et al. [22] used a population approach based on the simplest version of the Baccam models [1] to incorporate the drug effect of oseltamivir, and fitted the model to viral titer data from 208 subjects from four oseltamivir studies. Other possible drug effects with different MoAs as well as their combination with oseltamivir were also tested and compared, providing valuable insights to identify effective anti-influenza A agents. Another related work by Hadjichrysanthou et al. [16] further simplified the same viral kinetics model to only two compartments (i.e., susceptible cells and free infectious virus particles) and fitted to two different datasets from oseltamivir and zanamivir experimental studies. A series of morbidity and viral growth-related measures were then derived, and their key determinants were analyzed, as a “first step” to aid in the design of clinical trials of candidate therapies.

Designing clinical trials for patients with severe influenza is highly challenging. To optimize dose selection and study design in the clinical development of a novel human monoclonal antibody to treat hospitalized patients with influenza A infection, Patel and colleagues [33,34,35] proposed a comprehensive PK/PD modeling approach. More details on how influenza viral kinetics models have been used to support clinical development, especially for severe influenza infections, are presented in the next section.

With the advances in computational capacities, some recent work has utilized multi-scale mathematical modeling approaches. In the work by Heldt et al. [20], an intracellular level of modeling was used to describe relevant dynamics within an infected cell and was coupled with the extracellular level of infection. It comprised the growth and death of uninfected cells, their infection by free virions, the production of virus by infected cells, viral clearance/degradation, virus-induced apoptosis, and the lysis of apoptotic cells. This model provides an ideal platform to include further complexity into the intracellular level. Going in the other direction, the work by Kamal et al. [23] developed a modular interdisciplinary platform to investigate the economic impact of oseltamivir treatment. It consisted of a pharmacology module with PK/PD components, an epidemiological module within which drug effect on transmissibility was linked to viral shedding duration, and a health economics module, the decision analytic tool. A related work by Fidler et al. [14] used agent-based modeling that created a multi-scale influenza model connecting pharmacology, viral kinetics, and epidemiology, from tissue to population scales.

Finally, quite a few recent publications have at least partly focused on practical challenges or technical aspects of influenza viral kinetics modeling. In particular, Beauchemin and Handel [3] investigated modeling of influenza within an individual host or a cell culture, together with their kinetic parameters and values obtained, and showed a large variation resulted from different experimental assays, hosts, and IAV strains. On the other hand, Dobrovolny et al. [11] examined published models incorporating immune responses constructed from different data using various strategies and assumptions, and found that no single model agreed completely with the variety of influenza viral kinetics response when various immune responses are suppressed. Challenges with model robustness and parameter identifiability associated with the above findings were emphasized by the review by Boianelli et al. [5], and the authors proposed a step-wise approach for model identification which they demonstrated with a case study with IAV infection including the immune response. Most recently, Patel et al. [36, 37] assessed the performance of a variety of estimation methods in modeling the kinetics of respiratory virus infections.

Applications in Drug Development

Influenza viral kinetic models have been used to support the clinical development and dose optimization of neuraminidase inhibitors and other emerging drug candidates. In 2002, Iyer et al. developed a model that related the effect of an oral neuraminidase inhibitor (RWJ-270201) to changes in influenza viral titer [21]. In this study, a population PK model was initially developed, which was used to provide drug exposure as input into the PD model. Antiviral efficacy was then estimated using a sigmoid E max model that was parameterized using cumulative drug area under the curve (AUC) at the dosing regimens investigated.

Models have also been developed to describe how oseltamivir treatment may lead to the emergence and spread of drug resistance [19]. Canini et al. extended this model by including oseltamivir PK, symptom dynamics, and inter-individual variability and suggested that the time of initiating prophylactic antiviral therapy was a primary factor in the emergence of viral drug resistance [8]. In their model, drug-sensitivity or resistance was represented using a simple E max model parameterized as a function of oseltamivir carboxylate (OC) concentration.

Similar structural PK/PD models have since been used to predict oseltamivir effectiveness, based on the MoA for inhibition of viral production [6, 22, 33, 34]. Kamal et al. utilized their model to illustrate that early treatment with oseltamivir (at 0.5 to 1 days post-infection) is likely to maximize the decrease in the duration of viral shedding [22]. In addition, this model demonstrated the potential for a moderate additive effect when combining oseltamivir with drugs that inhibits influenza infectivity at a different phase in the viral life-cycle. An integrated disease model has been developed by Patel et al. that relates the oseltamivir concentration-effect on influenza VK, host cytokine modulation, and symptom measures [33, 34]. Recently, Boianelli et al. simulated oseltamivir treatment strategies following co-infection with influenza and bacterial pathogens [6]. This model suggested that doubling the standard 75 mg curative regimen may result in improved antiviral and antibacterial efficacy. Taken together, the above mechanistic models have served to provide improved guidance in the anti-influenza chemotherapy of neuraminidase inhibitors in various clinical scenarios.

In addition to the neuraminidase inhibitors, several monoclonal antibodies are currently in phase 1 or phase 2 clinical development for the treatment of influenza virus infection [31]. Recently, we developed a semi-quantitative PK/PD model that described MHAA4549A serum and nasal PK, and the corresponding impact on influenza viral load [33, 34]. Simulations from this model suggested that a dose higher than 3600 mg may have better efficacy in hospitalized patients. More recently, we constructed a similar mechanistic model that linked influenza dynamics to the PK/PD of VIS410, a novel human IgG1 monoclonal antibody with broad antiviral activity [36, 37]. This model characterized the impact of nasal PK on viral load and was used to support dose selection for future clinical development across various populations of interest.

Developing viral kinetic models relies on estimating key parameters that describe the influenza life-cycle [38]. In addition, the determinants of antiviral efficacy must be defined [39]. Typically, the PD effect has been incorporated using a general inhibitory (INH) function (Eq. 1) that based on the drug MoA, influences a change in the viral kinetics [6, 8, 19, 22, 33, 34]. Equation 2 illustrates the antiviral inhibitory effect using a representative example for inhibition of viral production, p:

$$ \mathrm{INH}\ (t)=\frac{E\max \times {C}^{\gamma }(t)}{{EC_{50}}^{\gamma }+{C}^{\gamma }\ (t)} $$
(1)

where E max is the maximal effect, C is the concentration of antiviral drug, EC50 is the drug concentration that produces half-maximal inhibition of virus, and γ is a Hill coefficient.

$$ p=p\times \left(1-\mathrm{INH}\right) $$
(2)

In addition to predicting drug efficacy, several models have been described to incorporate the influence of viral titer on the time needed for patients to resolve all disease symptoms and cease viral shedding [7, 33, 34, 39].

Typically, human challenge studies involve artificially inoculating healthy adult volunteers intra-nasally with an attenuated strain of influenza virus [2]. The viral time course is then evaluated following placebo or defined regimens of investigational drug treatment over a period of 7–8 days after admission to quarantine. Influenza viral load is measured using nasopharyngeal sampling, usually in duplicate from each nostril (https://www.cdc.gov/flu/). This procedure is invasive and can produce large estimates of inter- and intra-patient variability [35]. An advantage of human challenge studies is that they are conducted in a controlled environment where the time of influenza virus infection is known and recorded [9]. In contrast, the exact time of infection is unknown in patients naturally infected by the virus and cannot be predicted in models that describe human challenge data. To the best of our knowledge, no published models allow extrapolation from the controlled human challenge setting to patients naturally infected by the virus.

One potential limitation is that human challenge studies are often designed to administer drug dosing regimens that illustrate supra-therapeutic effect. As a consequence, the drug concentrations achievement may often far exceed the true potency, thereby precluding credible estimation of the EC50 value [35]. Model-based clinical trial design of human challenge studies is needed to enhance estimating viral life-cycle and associated PK/PD parameters [35,36,37]. However, a paucity of optimal design strategies supports the clinical development of novel and existing anti-influenza therapies. This information is valuable and can enhance future dosing recommendations for the prophylaxis and treatment of influenza infection.

Extensions of Viral Kinetic Models and Future Directions

Future opportunities for enhancing existing models of influenza viral dynamics, which will further increase their utility, include both making them more mechanistic in nature, as well as extending the use of these models into adjacent areas, including the use of multi-scale approaches.

Utilizing recent advances in quantitative systems pharmacology should allow for more detailed and mechanistic approaches in the future to capture intracellular components of the viral life cycle, including the incorporation of resistance to antivirals. This would allow for an enhanced understanding of potential therapeutic targets and improve the ability to leverage combination treatment options by testing in silico the impact of multiple mechanisms of action to identify the most promising and effective drug combinations.

Current viral dynamic models are limited by their lack of consideration of the host immune response. The host immune response to influenza is a critical component of the clinical course of the infection, determining why some patients have modest symptoms, whereas others may require hospitalization with more severe disease despite infection with the same virus. While some attempts have been made to incorporate cytokines such as interleukin 6 (IL-6) or IFN, there is generally high variability in these cytokine responses, which have not been adequately linked to clinical symptoms.

Exciting and emerging areas of viral dynamic modeling research involves the utilization of multi-scale models to link adjacent areas together which have not traditionally been done. Very sophisticated models of influenza epidemiology have been created, using both differential equation based-methods, and more recently, agent-based modeling methods ([13, 15, 28], and [24]). These models describe the transmission of influenza through a population of interest, informing at a macro level, transmission and spread of influenza, impact of control measures such as the use of masks, school closures, and vaccines. While extremely powerful, these epidemiology models typically make simplistic assumptions regarding variability in viral dynamics, antiviral activity, including the impact of dose, adherence, and emergence of resistance. The ability to make these more sophisticated and realistic viral dynamic models as inputs into models of epidemiology offer the opportunity to make better decisions regarding how influenza is managed on a global scale.

Recently, a demonstration of linking viral dynamic and epidemiology models has been described [23]. An additional example, evaluating the potential use of a monoclonal antibody with a long systemic half-life as an alternative to a prophylactic vaccine was evaluated using a similar methodology [43]. This manuscript concluded that a single dose of a broad spectrum monoclonal antibody as an alternative to a vaccine could have a significant impact from a public health perspective.

The introduction of agent-based models of influenza epidemiology allows one to model representative individuals in a population of interest, be it a city, state, or country ([12, 25, 30], and [29]). Each individual in the population is an ‘agent,’ with its own demographic and behavioral characteristics. The agents in the model interact with one another and are geographically mapped within their environment, thereby commuting to work or school and coming into contact with other agents where they may become exposed to influenza. Some of these models have very sophisticated geographical mapping, complete with robust commuter traffic, population densities, and the US air traffic system. Based on severity of influenza symptoms, agents may choose to stay home from work or school, which impacts the spread of the virus as these individuals are not coming into contact with other agents. The integration of more detailed multi-scale models allows the incorporation of viral dynamic models, where each agent or individual has their own specific viral dynamic, population PK, drug or vaccine adherence, and antiviral PK/PD model parameters governing their individual influenza infection ‘experience.’

These multi-scale models can be further enhanced to incorporate health economics, whereby the output of the epidemiology models can serve as the input to health economic and pharmacoeconomic models. The result is a powerful and modular tool allowing for the evaluation of many potential interventions in the management of influenza globally. The decisions which these models inform are of critical importance, as many of the questions regarding how to optimize control, treatment, and care of influenza cannot be practically tested in clinical trials or other experimental protocols. These models can be used to inform decisions regarding stockpiling of antivirals, to evaluate the comparative effectiveness (including cost effectiveness) of new therapies compared to existing treatments, and to evaluate real-world trade-offs between different interventions including vaccine strategies, masks, rapid diagnostics, school closures, and travel restrictions.

The above approaches have generally been pioneered in the area of influenza. However, there are many additional opportunities to apply these concepts to other existing and emerging pathogens, including respiratory syncytial virus (RSV), Ebola, smallpox, tuberculosis, and malaria.

As major advances in computational power and mathematical modeling methods continue to be made, the emerging ability to link sophisticated models across different scales of space and time will significantly improve the ability to obtain a more unified characterization of the different components and mechanisms governing the behavior and causality of relationships which determine the outcomes of influenza infection at the individual and public health levels.

Conclusions

  • Viral kinetic models have been an important tool for understanding and treating influenza since the 1970s and continue to be refined and expanded to the present day

  • Influenza viral kinetic models have been used to support the clinical development and dose optimization of neuraminidase inhibitors and other emerging drug candidates.

  • The utility of influenza viral kinetic models may be further enhanced through incorporation of greater mechanistic detail and interfacing VK models with epidemiological and pharmacoeconomic models