Quantitative viral dynamics: Methods for parameter estimation

Abstract

Fitting mathematical models of viral dynamics to serial, quantitative viral load data provides inferences on the mechanisms in virus infection. This process can reveal the speed and magnitude of viral replication, cell proliferation and death, immune responses, and/or treatment efficacy. Viral dynamics modeling involves developing conceptual models, translating them into equations, and applying the appropriate statistical tools to determine the optimal parameters such that the model recapitulates observations from human and animal infections. In this review, we outline the theoretical foundations needed to understand model fitting, parameter estimation, and what it means to achieve a good fit. We provide examples and explain the strengths and limitations of three commonly used model fitting approaches: individual fitting, population mixed effects fitting, and feature fitting. We briefly review fitting algorithms and highlight powerful available computer software packages that can be used for fitting and parameter estimation. We discuss different model types, parameter identifiability, and how future modeling efforts can leverage advances in multi-dimensional data. Finally, we conclude with simple guidelines for choosing the best approach based on available data and scientific questions.

1. Introduction

During viral infections, viral load and immune response data regularly show complex, oscillatory, and/or nonlinear behavior. Therefore, mathematical models of virus dynamics expressed as systems of ordinary differential equations (ODEs) provide an ideal framework to understand complex viral and immune system interactions, determine the strength and timing of immune responses in clearing infections, and predict how therapeutic interventions impact disease progression (Iwami et al., 2012a, 2012b; Ribeiro et al., 2010; Vaidya et al., 2021; Hernandez-Vargas and Velasco-Hernandez, 2020; Duke et al., 2021a; Du and Yuan, 2020; Nampala et al., 2018; Reeves et al., 2021; Perelson and Nelson, 1999; Nowak and May 1993; Goyal et al., 2022a; Smith, 2018). Viral dynamics models typically include state variables that capture the quantity of cells or viruses over time, and model parameters describe how often events like infection, viral clearance, or viral- or immune-mediated killing of infected cells occur. Solving the system of equations (typically computationally) generates trajectories of the state variables, a subset of which can be quantitatively compared to clinical and experimental data. These data often consist of serial viral loads but may also include the number of uninfected and infected cells, as well as the immune response over the course of infection. Model fitting is the act of tuning parameters, i.e., adjusting the numerical values that govern the magnitudes and rates of each model process. Proper model fitting results in parameter estimation, the identification of parameter values that make the model output closely resemble the data, thus quantifying the magnitudes and/or rates of mechanistic processes underlying viral infection.

Integrating empirical viral load data with mathematical modeling has contributed to effective treatment strategies and public health interventions (Jenner et al., 2020). Pioneering studies in the 1990s that fitted simple mathematical models of HIV dynamics to clinical data demonstrated that HIV replication is continuous and highly productive, with billions of virions produced and cleared daily (Ho et al., 1995; Wei et al., 1995; Perelson et al., 1996). These models revealed that HIV is removed rapidly from the bloodstream (~30 min) and that HIV-infected cells have a short half-life (~2 days). In the context of the known high error rate of the HIV reverse transcriptase, these studies accurately predicted that the use of multiple agents targeting different steps in the HIV replication cycle would be necessary to avoid drug resistance (Althaus et al., 2009; Cardozo-Ojeda et al., 2021). Modeling has also precisely identified rates governing the HIV latent reservoir, its persistence despite prolonged antiretroviral therapy (ART), the probability and dynamics of viral rebound after analytical treatment interruption, and many curative strategies (van Dorp et al., 2020; Conway et al., 2021; White et al., 2022; McMyn et al., 2023; Reeves et al., 2023a, 2023b).

For hepatitis C virus (HCV) and hepatitis B virus (HBV) infections (Reinharz et al., 2019), mathematical modeling led to optimization of interferon-based therapies and identification of therapeutic non-responders (Neumann et al., 1998; Guedj et al., 2010). Further, a proliferation of modeling of SARS-CoV-2 data elucidated a critical role for early innate immune responses in determining peak viral load (Kim et al., 2021), highlighted the critical importance of pre-symptomatic peak viral load in transmission and allowing super spreader events (Goyal et al., 2021), explained heterogeneity of viral loads across hundreds of infected individuals according to viral variant and vaccine status (Iyaniwura et al., 2024; Perelson and Ke, 2021; Owens et al., 2024), demonstrated kinetics of drug resistant variants during neutralizing antibody therapy (Phan et al., 2024), and explained viral rebound following antiviral treatment (Esmaeili et al., 2024).

As the field of modeling viral dynamics has matured, the notion of how to quantitatively determine if a model fits well to a dataset has been refined substantially. Here, after a brief description of the modeling process (Section 2.1) (Shabman et al., 2024), the basic mathematical model of viral dynamics (Section 2.2), and the notation used herein (Section 2.3), we illustrate three approaches used for parameter estimation (Section 3). We describe the theory behind each method and provide examples of its use in viral dynamics. We then describe various objective functions (which are used to quantitatively score model fit to data, Section 4.1), discuss relevant computational algorithms and recommend software packages currently available for model fitting (Section 4.2). Finally, we discuss common challenges and considerations for choosing the optimal model fitting strategy, with a specific consideration of forthcoming multi-dimensional data types (Section 5).

2. Modeling, fitting, and parameter estimation

2.1. The modeling cycle

The ideal cycle of mathematical modeling proceeds from a specific scientific question and experimental data to modeling and eventually back to experiments for verification (Fig. 1). The steps that modelers often follow are (1) identify the scientific question, (2) develop and analyze competing mathematical models, and (3) use the model to answer the scientific question, and/or predict future behavior of the biological system. A key component of the model development step is determining model parameter values through model fitting, which should be accompanied by performing a literature search and understanding the relevant experimental data. Though not always possible, ideal modeling completes the cycle with a validation step, where model-generated hypotheses or predictions are validated against additional data not used for fitting. Model selection, which is not covered in depth here, is an increasingly crucial part of the modeling cycle (for in depth coverage, see the influential text by Burnham and Anderson (2004a)). Briefly, a pool of candidate models is developed that use slightly different equations to encode various competing mechanistic hypotheses. The optimal model structure for the data is chosen based on data fit and parsimony (Burnham and Anderson, 2004b). Importantly, rigorous model selection requires reproducible and accurate model fitting and thus relies on the considerations in the present work.

Fig. 1. Steps to use mathematical modeling to address biological questions of interest.

(1) Beginning with identifying the question of interest about virus infection, investigators (2) develop mathematical models based on biological hypotheses. The model that best represents the biological system is then selected (A) and fitted to data by choosing a fitting approach and performing parameter estimation with an appropriate dataset. (B) After a thorough literature search, parameter estimation methods, such as individual-level, population level, or fitting to population features can be used to determine the rates and magnitudes of processes underlying an infection. (C) The model fitted to data is used to make testable projections about infection outcomes under different conditions. Interpreting model results and drawing conclusions lead to (3) answering the original question and formulating new biological insights.

2.2. The basic model of viral dynamics

Throughout this review, we will illustrate concepts using the canonical viral dynamics model for persistent infection (Perelson and Nelson, 1999; Nowak and May 1993; Perelson et al., 1996; Nowak et al., 1996). Viral dynamics models are often usefully visualized as cartoons in which each population of interest is shown, and arrows connecting the populations represent the transition processes that occur (Fig. 2A). The canonical viral dynamics model is expressed mathematically using ordinary differential equations that characterize the rate of change in each population (Fig. 2B–Equation (1)). In the canonical model, we consider the population sizes of three state variables: uninfected (i.e., susceptible) cells $(U)$, which become infected cells $(I)$ when infected by virions $(V)$. Model parameters govern the rate of each process occurring. Here, uninfected cells are produced with rate $\lambda$ and die naturally at rate ${\delta }_U$; when virions encounter uninfected cells, infection occurs with rate $\beta$, and infected cells die with rate ${\delta }_I$ (generally much faster than natural death); infected cells produce virus with rate $p$, and virus is cleared with rate $c$.

Fig. 2. Basic viral dynamics model.

(A) In the basic model of viral infection, uninfected cells susceptible to infection, $U$, are infected by virus, $V$, with rate constant $\beta$. Uninfected cells are assumed to be produced by a source at rate $\lambda$ and to die at per capita rate ${\delta }_U$. Infected cells, $I$, produce virus at rate $p$ and die at rate ${\delta }_i$. Free virus particles, $V$, are cleared at rate $c$. (B) This model can be encoded as a system of three ordinary differential equations describing the rates of change of $U,I$ and $V$ over time.

$$\begin{gathered} \frac{dU}{dt}=\lambda -{\delta }_{U}U-\beta UV \\ \frac{dI}{dt}=\beta UV-{\delta }_{I}I \\ \frac{dV}{dt}=pI-cV \end{gathered}$$ (Eq. 1)

Most modern viral dynamic models include variations to the basic model to account for additional complexities of viral infections. Many variations of this model are able to explain observed data from different viral infections. Inclusion of the term $-\beta UV$ in the $\frac{dV}{dt}$ equation (representing the loss of free viral particles after cell infection) is more biologically precise, but generally the contribution of this term is considered negligible and it is omitted from ODE models. Some common additions to this model include another state variable $E$ to describe the effector response (i.e., humoral and/or cellular immune responses) with additional terms to represent adaptive immune-mediated mechanisms such as lysis of infected cells, clearance of free viruses, or antibody neutralization (Nowak et al., 1996; Khoury et al., 2021; Hull-Nye et al., 2023); inclusion of an eclipse phase (Baccam et al., 2006; Canini and Carrat, 2011); inhibition of viral replication (Smith, 2018; Baccam et al., 2006; Canini and Carrat, 2011; Beauchemin and Handel, 2011) or infectivity due to therapy (Goyal et al., 2022a); modeling innate immunity via density-dependent removal of infected cells or inclusion of a refractory cell compartment (Perelson and Ke, 2021; Esmaeili et al., 2024); and reactivation of virus from latency (Schiffer et al., 2011, 2013a; Byrne et al., 2021). Additions should always be considered based on the scientific question and available data.

Importantly, this canonical model is based on chemical reaction theory. The assumptions are logical but not necessarily verified fully in viral systems. For instance, mechanisms typically assumed to be encoded by parameter constants might be allowed to vary over time, or instead be described by additional state variables.

Finally, in the ideal mathematical case, a model can be solved analytically (yielding, for example, viral load as a function of time, $V(t)$), which can be immediately fitted to the data using linear regression (Perelson and Nelson, 1999; Perelson et al., 1996). Also, studying the underlying structure of models can illuminate various steady states even when parameters are not known precisely (Conway and Perelson, 2015). However, because most models are complex and include additional nonlinear terms compared to the canonical model, numerical integration is more commonly used to solve the equations and evaluate the solutions at the observation time points required.

2.3. Notation

In this review, we will use vector notation to describe the general list of state variables, $\mathit{X}(t)$ at some time $t$, such that for Eq. (1), $X(t)=[U(t),I(t),V(t)]$. We denote a model abstractly as $M[\mathit{X}(t),\mathit{\theta },t]$, indicating a set of rules that take the state variable levels at any time $t$ and relate them together. Boldface indicates the model is itself a system with multiple equations. We will use the notation $x(t)$ to represent one state variable (such as viral load, $V(t)$). The model also takes as input a vector, $\theta$, of parameter values. In the case of the canonical system above, $\theta =\left[\lambda ,{\delta }_U,\beta ,{\delta }_I\right.,p,c]$. The output of the model is the change in each state variable $[dU/dt,dI/dt,dV/dt]$.

Common symbols and notation are summarized in Table 1, and a glossary of key terms, which are italicized as they arise throughout the review, is provided in Box 1.

Table 1.

Description of mathematical symbols used.

Mathematical symbol	Description
$t$	Time
$U(t)$	Uninfected cells
$I(t)$	Infected cells
$V(t)$	Viral load
$\beta$	Infectivity rate
$\lambda$	Production rate of uninfected cells
${\delta }_U$	Death rate of uninfected cells
${\delta }_I$	Death rate of infected cells
$p$	Virus production rate by infected cells
$c$	Virus clearance rate
$x(t)$	One state variable, typically viral load $(V(t))$
$\mathit{X}(t)$	A vector of state variables (e.g. $[U(t),I(t),V(t)]$)
$\mathit{\theta }$	Vector of parameter values (e.g. $\left[\beta ,\lambda ,{\delta }_U,{\delta }_I,p,c\right]$)
$M[\mathit{X}(t),\mathit{\theta },t]$	Mathematical model, the set of equations describing the state variables over time
$N$	Total number of individuals in a population or repetitions in an experiment
$i$	Subscript representing the $i$-th individual
$j$	Subscript representing the $j$-th time point
$k$	Subscript representing the $k$-th parameter in the parameter set
$T_i$	Total number of time points for individual $i$
$t_j$	The $j$-th time point
${\mathit{D}}_{\mathit{i}}$	Vector of longitudinal data for individual $i$
$D_{ij}$	Data point from $i$-th individual at $j$-th time point $t_j$
$\mathit{ϵ}(\mathit{X}(t))$	Error model which can be a function of state variables, time, or both
${\mathit{\theta }}_{\mathbf{\text{pop}}}$	Population parameter set (mean of population parameter distributions)
${\mathit{\eta }}_i$	Individual $i$-th random effect for each parameter in the parameter set
pdf	Probability density function
$𝒩$ (mean, std. dev.)	Normal distribution
$l$	Total number of parameters in the model
$\mathbf{\Omega }$	An $l\times l$ matrix of covariates with diagonal elements representing the variance of each parameter and non-diagonal terms representing the correlation between the parameters
$\sqrt{{\Omega }_{kk}}$	Standard deviation of the random effect of $k$-th parameter
$\sigma$	Standard deviation of the measurement error $\mathit{ϵ}$
${\mathit{L}}_{ij}$	The likelihood of mathematical model generating data point $D_{ij}$ given parameter set ${\theta }_i$
${\mathit{L}}_i$	Total likelihood function for individual $i$
$P\left({\mathit{\theta }}_i\right)$	Prior distribution of parameters for individual $i$
$P\left({\mathit{\theta }}_i\mid {\mathit{D}}_i\right)$	Posterior distribution of parameters for individual $i$
${\mathit{D}}_{\mathit{\text{i cens}}}$	The vector of censored data for individual $i$
${\mathit{L}}_{\mathit{\text{i cens}}}$	Likelihood of complete data (observed and unobserved (censored) data)
$\varphi$	A prespecified tolerance measure for the distance between the modeled and observed features in the ABC (Approximate Bayesian Computation) approach

Box 1. Glossary of key terminologies used.

Censored Data – Data (e.g., measurements of viral load) that are below or above a certain level of quantification such that their values cannot be measured or reported.

Convergence – When parameter estimates in each round of an iterative process get increasingly closer to a set of parameter values that optimize the objective function.

Covariates – Independent variables (or features like sex, weight, etc.) that can explain some of the variability in the observed data and can be used to guide the estimation of parameter distributions.

Eclipse Phase – Phase after cell infection before the production of new virions begins.

Goodness of Fit – A statistical measure quantifying how well a model’s output fits the observed data.

Imputation – Replacement of missing values in a dataset with estimated values.

Longitudinal Data – Data collected through repeated observations of the same variable over time, e.g., viral load from nasal swabs collected throughout a viral infection.

Mixed Effects – Combining population-wide patterns (i.e., fixed effects) with individual variability (i.e., random effects) to simultaneously estimate population distributions of parameters and individualized parameter values.

Model Parameters – The rates governing the biological processes associated with the state variables in the dynamical system.

Model Selection – The process of selecting the most suitable mathematical model from a group of potential models to effectively explain or predict a dataset.

Non-convex – A curve with multiple peaks (maxima) and valleys (minima).

Objective Function – A mathematical formula that specifies the goal of an optimization problem, usually aiming to minimize the differences between simulated and observed data.

Ordinary Differential Equations – Algebraic relations involving derivatives, or rates of change, of one or more functions with respect to one independent variable (usually, time).

Parameter Distribution – The distribution that describes the possible values of parameters in a population and the likelihood of each value occurring for a given dataset.

Parsimony – Principle of selecting a model that explains the data with the fewest possible parameters, favoring simpler models over complex ones that achieve a similar level of fit to the data.

Posterior Distribution – Conditional parameter distributions estimated by modifying the prior distributions based on the likelihood of the model generating the new data.

Parameter Estimation – Finding the best values of the model parameters that recapitulate the observed data by optimizing the objective function.

Parameter Identifiability – Analysis of how accurately model parameters can be estimated based on the associated experimental data.

Parameter Set – A collection, or vector, of the model parameter values.

Parameter Values – Numerical values assigned to the rates governing the dynamical system.

Practical Identifiability – Analysis on whether unknown model parameters can be estimated with reasonable accuracy and confidence from real, noisy experimental data.

Prior Distribution – Assumed parameter distributions based on prior knowledge of the biological system.

State Variables – Dependent variables in the viral dynamics model describing the behavior of the components of the system over time, e.g., uninfected cells, infected cells, and viral load.

Structural Identifiability – Analysis on whether unique values of unknown model parameters can be estimated from error-free input-output data based on the model structure.

Viral Dynamics Models – Mathematical models describing the behavior of viral infections within a host organism.

3. Approaches to model fitting

Model fitting is the process whereby the numerical values of each parameter (e.g., the rates governing the viral dynamics model) are adjusted until model output matches the data. Depending on the situation, different model fitting approaches have been used (Fig. 3). Individual fitting considers the data from each individual separately and attempts to match each individual’s viral load trajectory (Fig. 3A). Population nonlinear mixed effects (pNLME) approaches use data from multiple individuals to estimate population-level parameters that describe the average behavior of the population and the inter-individual variability, as well as estimate the best-fit parameters for each individual (Fig. 3B). For certain infections, longitudinal data preclude direct fitting due to the episodic nature of viral recrudescence or missing/noisy data. In this case, feature fitting of data across the population may be necessary (Fig. 3C). For instance, a model may be tasked with accurately recapitulating the distribution of levels and timing of peak viral loads across a cohort, rather than each individual’s trajectory.

Fig. 3.

Three approaches to fitting longitudinal data from multiple individuals. (A) In an individual-fitting approach, the optimal parameters, $\widehat{𝛉}$, are estimated for each individual one at a time such that evaluating the model with those parameters at the corresponding time points, $V\left(t_j,\widehat{𝛉}\right)$ for $j=1,\dots ,T$, best matches the data. There are no constraints on how different each parameter value can be across individuals (unless specified by a range), allowing maximum model flexibility. Summary statistics for each parameter can be constructed using the individual estimates. (B) Population nonlinear mixed effects modeling (pNLME) simultaneously estimates parameter values for each individual and population parameter distributions that collectively minimize the pNLME equation (Eq. (2)). Assuming a normal distribution for a given parameter, say the infectivity $\beta$, an individual $i$’s parameter estimate, ${\beta }_i$, is the sum of the population value, ${\beta }_{\text{pop}}$, and an individual “random effect”, ${\eta }_i$, that reflects its deviation from the average behavior, where ${\eta }_i$ is normally distributed with a mean of 0 and standard deviation of ${\omega }_{\beta }$. The plot on the right illustrates the population parameter distribution of $\beta$ given the observed data, $D$, as normally distributed with a mean of ${\beta }_{\text{pop}}$ and standard deviation of ${\omega }_{\beta }$. Assuming a structure for the underlying distribution governing how parameter values can vary across individuals can help enhance parameter identifiability in cases of limited data but may constrain parameter values to a narrow regime around the population value. (C) Fitting to population features entails first defining a set of quantitative features to target and calculating these values from data. Then an iterative algorithm like Approximate Bayesian Computation (ABC) can be used to identify a parameter regime under which the model produces simulated results similar to the target data. In this algorithm, the model is run using many parameter sets, ${\mathit{\theta }}^{(i)}$, and the quantitative metrics are calculated based on the resulting simulated data. Simulated data is compared with observed data for each target metric using a distance measure $\rho \left(D_{\text{sim}},D\right)$. Within some pre-defined tolerance, $\varphi$, a parameter set is accepted as hitting the target for that metric (e.g., ${\mathit{\theta }}^{(\mathit{1})}$). Parameter sets that miss the mark for too many metrics are rejected (e.g., ${\mathit{\theta }}^{(\mathit{2})}$). Over several runs, and possibly narrowing the tolerance, this process converges to a final parameter distribution for the population. Symbols appearing in the figure are defined in Table 1.

Parameter values are estimated by determining the set of model parameters that minimize the distance between the data and the model-generated output across the observation time points. Least squares method is often used, which minimizes the sum of squared errors (SSR). A more general procedure called maximum likelihood estimation (MLE), which maximizes a user-defined likelihood function, can also be used; both the sum of squared errors and the likelihood function are examples of objective functions, described in detail below. The “best fit” of a model to a dataset is determined by optimizing the objective function.

3.1. Individual-level fitting

In individual fitting, the optimization procedure is performed for each individual (or study participant) separately, giving rise to a set of model parameters that best fits the individual’s data (Fig. 4). The parameter sets derived from fitting each individual can then be collected and summarized into typical statistical distributions. Several characteristics that enhance how well the model can be fitted to the data include a high number of data points, a high sampling frequency, and robust experimental and laboratory protocols. Thus, strengths of the individual fitting approach include flexibility, computational simplicity, and transparency. A drawback of fitting individuals separately is that estimated parameters may vary by multiple orders of magnitude across individuals, which may not be biologically realistic but can occur even when other model parameters are fixed. Another is that all individual estimates are treated as equally valid. For these reasons, individual fitting is preferable for when population level fitting is not possible, when simplicity and transparency are needed, and for well-sampled longitudinal data from a small number of individuals.

Fig. 4. Example of parameter estimation for one individual.

(A) Simulated viral load curves produced using the model (gray lines) with different sets of parameters can be compared against longitudinal data (green points). The model simulation using optimal parameter values (red line) best matches the data (i.e., black bars measuring distance are smallest in total). (B) For any single parameter (here viral infectivity rate $\beta$), changing the value of this parameter will change the value of the objective, or scoring, function. Here we show the least-squares objective function (residual sum of squares, RSS) that measures the squared differences between the model solution at time $t_j$ (i.e., $M\left[x\left(t_j\right),\theta ,t_j\right]$) and the observed data at time $t_j$ (i.e., $D_j$) for time points $j=1\dots T$, i.e., the sum of squared black bars in panel (A). The RSS is minimized simultaneously over all parameters to determine the set of parameter values $\widehat{\mathit{\theta }}$ that best fits the observed data. It is possible that a value can admit a low RSS relative to the nearby values (e.g., ${\beta }^3$), but not achieve a full “global” minimum $(\widehat{\beta })$. Certain numerical algorithms are more robust to escaping such local minima and finding the global value. (C) The parameter values that produce the optimal simulation curve provide estimates of the rates governing the viral dynamics system. Statistics like 95% confidence intervals can also be constructed using the RSS framework.

Examples of individual-level fitting.

The earliest examples of estimating viral dynamic parameters with the canonical viral dynamics model took an individual-level fitting approach. For instance, individual-level fitting using data from 10 people who acquired HIV were used to estimate viral infectivity and suggested a potential adaptive immune response to early viremia (Stafford et al., 2000). A variant of this model was used on highly granular data from macaques inoculated with simian HIV (SHIV) to individually fit primary infection curves as well as rebound viremia after treatment cessation (Reeves et al., 2017). This model estimated that rebound viremia is blunted because a higher initial level of effector cells before rebound (vs. primary infection) helps control viral expansion. In another study, data from 47 people who acquired HIV were used to estimate the initial viral growth rate and HIV’s within-host basic reproduction number ($R_0$, or the typical number of infected cells generated by a single infected cell during early infection) (Ribeiro et al., 2010), yielding a mean $R_0$ of ~8, with interquartile range 4.9–11.

A characteristic and compelling example of individual fitting is an investigation of lentiviral infection dynamics and immune control in immunodeficient horses inoculated with equine infectious anemia virus (EIAV) (Schwartz et al., 2018). The study estimated five kinetic parameters of infection, using a rare dataset of frequent viral loads during early infection in seven horses with severe combined immunodeficiency. The estimates revealed that EIAV is at most mildly cytopathic to infected cells, and the rate at which infused antibodies neutralized the virus was at least 18 times greater than the virus clearance rate, providing the minimal efficacy of antibodies that successfully prevented infection. This study offers insight into the scale of protection needed by a potential antibody-eliciting vaccine or intervention.

Further studies using individual level fitting (among others) have investigated the kinetics and infectiousness of SARS-CoV-2 in non-human hosts (Heitzman-Breen and Ciupe, 2022; Vaidya et al., 2021; Gonçalves et al., 2021), vaccine-induced IgG and IgA against SARS-CoV-2 (Murphy et al., 2024), hepatitis C virus and the antiviral effects of interferon-alpha therapy (Neumann et al., 1998), decreased infectivity of simian immunodeficiency virus (SIV) during primary infection (Vaidya et al., 2010) and higher SIV susceptibility and viral loads in morphine-dependent animals (Vaidya et al., 2016), quantification of antibody escape kinetics of EIAV infection in horses (Schwartz et al., 2015a), quantification of innate and adaptive immune responses in HIV (Wu et al., 2008; Conway and Ribeiro, 2018), the effectiveness of HIV antiviral treatments (Conway and Perelson, 2015; Perelson et al., 1997), and viral kinetics of influenza infection and treatments (Smith, 2018; Handel et al., 2010; Miao et al., 2010; Dobrovolny et al., 2011, 2013; Hernandez-Vargas et al., 2014; Smith et al., 2018; Hooker and Ganusov, 2021). These studies and this approach continue to provide valuable insight into infection dynamics.

3.2. Population nonlinear mixed effects (pNLME) fitting

When sufficient high-quality data are available from multiple individuals, it can be preferable to make some assumptions about how individual parameters behave relative to a “population” distribution with a characteristic average and variance, and that variability in certain parameter values drives observed heterogeneity in viral load data. Population non-linear mixed effects models (pNLME) identify population-wide patterns by simultaneously estimating a parameter distribution to describe the full population, and a set of parameter values for each individual trajectory (Luo et al., 2022). These models include three main components: a mechanistic model (e.g., the model $\mathit{M}\left[\mathit{X}\left(t_j\right),{\mathit{\theta }}_{\mathit{i}},t\right]$ described in section 2.3), a model for the population parameter distributions (e.g., the distribution of viral production rates is log-normal), and a measurement error model (e.g., the error is normally distributed). We denote the measurement error $\mathit{ϵ}(\mathit{X}(t))$, indicating that errors can depend on the state variables, which themselves are a function of time. Thus, in a pNLME model, the equation representing experimental data (e.g., viral load) from the $i$-th individual can be written

$$D_i(t)=M\left[x\left(t_j\right),{\mathit{\theta }}_i,t\right]+\mathit{ϵ}(x(t))$$ (Eq. 2)

Importantly, the structure of the mechanistic model and the error model is the same for all individuals, but the parameter values, $\mathit{\theta }$, necessarily differ for each individual. Thus, many of the strengths of the individual fitting approach are shared by the pNLME approach.

Then, the pNLME model mathematically incorporates inter-individual variability by relating the population mean parameter to the individual parameters following each parameter’s assumed distribution. For instance, if all the parameters are normally distributed, the population parameter set ${\mathit{\theta }}_{\mathbf{\text{pop}}}$ is the mean, and individual values are normally distributed around this mean. When we fit individual trajectories then we adjust the “random effects” such that for each parameter in the vector, we have ${\mathit{\theta }}_{\mathit{i}}={\mathit{\theta }}_{\mathbf{\text{pop}}}+{\mathit{\eta }}_{\mathit{i}}$ where ${\mathit{\eta }}_{\mathit{i}}~𝒩(0,𝛀)$ where $𝛀$ is the $l\times l$ covariance matrix ($l$ is the number of parameters) reflecting the variance on each parameter as well as any cross-correlations among parameters. Thus $\sqrt{{\Omega }_{kk}}$ gives the standard deviation of the $k$-th parameter and if the parameter varies among individuals, this value will be large. Some viral dynamics parameters may span several orders of magnitude (and are naturally only positive values) such that using log-normal distributions are commonly used. Measurement error $\mathit{ϵ}$ is commonly assumed to follow a normal distribution that is constant for all values of the state variables over all time with standard deviation ${\sigma }^2$, i.e., $\mathit{ϵ}(\mathit{X}(t))~𝒩\left(0,{\sigma }^2\right)$. However, another common choice is that measurement error is proportional to the value of viral load, or that error is constant for log₁₀ viral load (Reeves et al., 2021; Lavielle, 2014; Lindstrom and Bates, 1990; Clairon et al., 2024).

An advantage of pNLME is that parameter values can be naturally constrained into biologically plausible distributions. The population approach also allows individuals with more frequent sampling to help fit individuals with lower sampling (“borrowing strength”). If there are different subgroups in the data which are already hypothesized to have distinct parameters, covariates are easily implemented in this approach. The drawbacks are statistical and computational complexity, and that certain parameter values may be “shrunk” to the mean of the population distribution when they have higher heterogeneity in reality.

Examples of pNLME fitting.

One recent example of pNLME is a model to understand the drivers of substantial heterogeneity in observed SARS-CoV-2 viral trajectories according to viral expansion slope, peak viral load, viral clearance kinetics, viral rebound, and duration of shedding. Capitalizing on the availability of densely sampled viral trajectories from 1510 individuals including critical early timepoints, Owens et al. used pNLME to test multiple competing models for SARS-CoV-2 infection for fit to the data (Owens et al., 2024). The optimal model suggested an early and intense innate response which tended to correlate with the timing and effectiveness of subsequent acquired responses which cleared the virus. Model output explained why certain individuals had short duration, low viral load shedding while others shed for weeks at higher viral loads. pNLME was the optimal approach for this dataset because it allowed for comparative model testing and generated realistic distributions of multiple unknown parameters which could then be assessed for their impact on infection outcome. Esmaeili et al. then used a pNLME approach to fit the same model to individualized data from a nirmatrelvir/ritonavir clinical trial. Model analysis explained a mechanism of post-treatment viral rebound and suggested that the optimal treatment regimen to limit viral rebound is extending treatment duration to 10 days (Esmaeili et al., 2024).

Further studies using pNLME fitting (Lavielle, 2014; Lindstrom and Bates, 1990) (among others) have estimated mortality in SARS-CoV-2 viral kinetics (Néant et al., 2021), viral resistance (Phan et al., 2024) and elimination (Goyal et al., 2020) to treatment, reduction of viral production due to remdesivir (Lingas et al., 2022), treatment effects of nirmatrelvir plus ritonavir (Hammond et al., 2022), super-spreading events of SARS-CoV-2 and influenza infections (Goyal et al., 2021), HIV viral rebound (Desikan et al., 2020), primary infection of HIV (Adams et al., 2005) in humans (Mdluli et al., 2022; Reeves et al., 2023c; Tettamanti Boshier et al., 2022), and the effects of influenza antiviral treatment of oseltamivir carboxylate in ferrets (Reddy et al., 2015).

3.3. Feature fitting

There are many reasons why longitudinal fitting to replicate individuals may not be possible. For instance, certain viral infections such as herpes simplex virus-2 (HSV-2) and CMV are characterized by stochastic mucosal shedding episodes (Schiffer et al., 2010; Mayer et al., 2017; Duke et al., 2021b). Individualized data can lack repeatable patterns and/or sampling frequencies can be so sparse or heterogeneous that longitudinal fitting would introduce artifacts rather than describe biological rules. There may be measurements of multiple different types such that a more complex objective function or likelihood is necessary to combine data. In this case, feature fitting (Fig. 3C) can be a useful framework.

Features might include the timing and magnitude of peak viral load, or the time when a particular immune response becomes detectable. Some examples of feature fitting pertain to human herpes viruses that have stochastic reactivation patterns with complex viral dynamics (Duke et al., 2021a; Byrne et al., 2021; Krantz et al., 2024). HSV-2 genital shedding episodes often have multiple erratic peaks, which exceed the complexity of single rebound trajectories that were observed and successfully modeled for SARS-CoV-2 (Schiffer et al., 2011, 2013a). Feature fitting has been used to link evolutionary and viral load data (Swan et al., 2022), and to deal with sparse data of multiple types (e.g., longitudinal measurements of paired viral load and cytokine expression profiles (Lim et al., 2023)).

An example of feature fitting.

Byrne et al. (2021) developed a stochastic model for the dynamics of Epstein Barr Virus (EBV), including viral load, infected cell levels, and immune response within the tonsillar epithelium. They estimated model parameter distributions by fitting to data from 85 Ugandan adults both with and without HIV co-infection using the Approximate Bayesian Computation (ABC, described below) algorithm with metrics including the frequency of positive swabs, median, maximum, and variance of detectable viral loads, and the number of viral load peaks during follow up. This approach helped identify that the rate of B cell reactivation and EBV-specific cytotoxic T cell proliferation and recruitment rate were affected by HIV-1 infection status. Moreover, HIV-1-coinfected individuals were found to have more actively infected tonsillar crypts, producing higher levels of EBV than HIV-1 uninfected individuals.

Further studies used feature-based fitting to characterize the dynamics of HIV RNA levels after starting potential antiviral treatment (Wu and Ding, 1999), estimate the time between infecting exposures and first positive tests during primary HIV infection (Reeves et al., 2021), characterize the evolution of HIV during primary infection driven by the intrinsic fitness of new viral variants (Swan et al., 2022), estimate parameters for maraviroc in HIV patients (Chan et al., 2011), and investigate associations between viral decay during ART and viral rebound after ART interruption (Gao et al., 2022).

4. Objective functions, algorithms and software

4.1. Objective functions

The goal of model fitting is to find the best set of parameters for each individual, $\widehat{{\theta }_i}$. To do so, we aim to optimize an objective function. Evaluating the objective function is a way to “score” how well the model fits the data or assess its accuracy. Here we describe several typical numerical objective functions.

4.1.1. Least squares

The least squares approach aims to minimize the sum of squared residuals (RSS) function, where the residuals are the differences between the model output and the data at each time point. For instance, a typical expression of RSS is the difference from the model output $x\left(t_j\right)$ representing viral load, when run with parameter list ${\mathit{\theta }}_i$ evaluated at each time point and compared to the observed viral load data $D_{ij}$ at each time point $t_j$, or

$${\text{RSS}}_i=\sum _j{\left[M\left[x\left(t_j\right),{\mathit{\theta }}_i,t_j\right]-D_{ij}\right]}^2$$ (Eq. 3)

where this RSS value is computed for each individual $i$ separately and thus a specific optimal parameter set is defined independently for each individual.

While the least squares methods are computationally efficient and relatively straightforward to implement, there are key limitations. First, the method relies upon the assumption of normally distributed residuals, or errors, between the model and observed data, with a mean of zero and constant variance, which may not always hold in real-world data and in particular for viral dynamics data. Second, the least squares method assumes that the observations are independent, i.e., there is no correlation between the residuals. This assumption is often violated when working with longitudinal data. Further, difficulties can arise in estimating parameters from data sampled over a broad time interval. Intelligent choices for parameter ranges and time intervals can help with optimization. Alternatively, sequential adjustment for series of increasing time intervals has been used to remedy this issue (Baker et al., 1997). A potentially more rigorous alternative to simple least squares is the generalized least squares method, where the residual matrix includes the correlations between observations (Adams et al., 2005).

4.1.2. Maximum likelihood estimation (MLE)

Maximum likelihood estimation (MLE) is a more general approach to parameter estimation. Unlike least squares, MLE can incorporate a variety of assumed error distributions, such as normal, exponential, or binomial distributions (Myung, 2003). This flexibility can be useful, for instance to deal with common realities such as censored data below a limit of detection, or assays that lead to greater uncertainty/noise at higher or lower viral loads. Unfortunately, MLE can be computationally demanding, especially for complex models with many unknown parameters or with incomplete or missing data. Convergence of MLE is sensitive to the initial choice of the parameters (Schwartz et al., 2015b). When data are incomplete, accurately evaluating the likelihood can be difficult, as it requires integration over possible values of the missing data or the use of imputation strategies. Despite these challenges, MLE remains a widely used method for parameter estimation due to its statistically rigorous framework (Lawson and Glenn, 2008).

Mathematically, the MLE framework defines a general function (the likelihood $L$) that quantifies the probability that a certain model with a certain parameter set ${\mathit{\theta }}_{\mathit{i}}$ generated a single data point:

$$L_{ij}=f\left(D_{ij}\mid {\mathit{\theta }}_{\mathit{i}}\right)$$ (Eq. 4)

The total likelihood function for the individual $i$ is then the product of the likelihoods for $T_i$ data points,

$$L_i=\prod _j^{T_i}{L}_{ij}.$$ (Eq. 5)

Taking the logarithm of the product in Equation (5) conveniently yields a summation. Thus, the log-likelihood for the data for individual $i$ (often expressed with lower case $l_i$) is

$$l_i=\text{log}\left(\prod _j^{T_i}{L}_{ij}\right)=\sum _j^{T_i}\text{log}\left(L_{ij}\right).$$ (Eq. 6)

The least squares objective function Equation (3) can be considered a specific case of the maximum likelihood method if we assume that the observed data are normally distributed. The model’s prediction for the optimum choice of parameters predicts the mean of the distribution. The likelihood function for individual $i$ can be expressed as

$$L_{ij}=\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{{\left(M\left[x\left(t_j\right),{\mathit{\theta }}_{\mathit{i}},t_j\right]-D_{ij}\right)}^2}{2{\sigma }^2}}$$ (Eq. 7)

where $\mathit{M}\left[x\left(t_j\right),{\mathit{\theta }}_{\mathit{i}},t_j\right]$ is the model’s solution to the state variable $x$ of the ODEs (which is often viral load, $V(t)$), with model parameters ${\theta }_i$ at time $t_j$. $D_{ij}$ is the observed data for time point $j$, and $\sigma$ is the standard deviation of the errors, assumed to be constant across all observations. The log-likelihood can be written as

$$\text{log}\left(L_{ij}\right)=-\frac{1}{2}\text{log}\left(2\pi {\sigma }^2\right)-\frac{{\left(M\left[x\left(t_j\right),{\mathit{\theta }}_{\mathit{i}},t_j\right]-D_{ij}\right)}^2}{2{\sigma }^2}.$$ (Eq. 8)

The total likelihood is the sum of all log-likelihoods over all the data points:

$$\begin{gathered} \sum _j^{T_i}\text{log}{L}_{ij}=\sum _j^{T_i}\left(-\frac{1}{2}\text{log}\left(2\pi {\sigma }^2\right)-\frac{{\left(M\left[x\left(t_j\right),{\mathit{\theta }}_{\mathit{i}},t_j\right]-D_{ij}\right)}^2}{2{\sigma }^2}\right) \\ =-\frac{T_i}{2}\text{log}\left(2\pi {\sigma }^2\right)-\frac{1}{2{\sigma }^2}\sum _i^n{\left(M\left[x\left(t_j\right),{\mathit{\theta }}_{\mathit{i}},t_j\right]-D_{ij}\right)}^2 \end{gathered}$$ (Eq. 9)

The first term in the above equation is constant and independent of the parameters, and therefore can be disregarded. The second term in Equation (9) can be written in terms of the RSS, which, when minimized, maximizes the likelihood.

Ultimately, what emerges from this approach is the parameter set that maximizes the likelihood, which is often expressed as

$$\widehat{{\mathit{\theta }}_{\mathit{i}}}=\underset{\theta }{\text{argmax}}{L}_i\left({\mathit{\theta }}_{\mathit{i}}\mid {\mathit{D}}_{\mathit{i}}\right),$$

where argmax notation indicates the argument (in this case the set of parameters ${\theta }_i$) that admits the maximum value of the function. Note in practice, the computational algorithms that search for the maximum determine an approximation of the true maximum likelihood parameter set.

4.1.3. Approximate Bayesian Computation (ABC)

Approximate Bayesian Computation (ABC) is a simulation-based inference algorithm increasingly used for model fitting (Pritchard et al., 1999). This method does not require defining or computing a likelihood function, which is useful for complex models with many unknown parameters (Minter and Retkute, 2019). The ABC method is also an applicable alternative to likelihood-based methods when working with incomplete experimental data, a scenario in which likelihood-based methods can be quite difficult to implement (Kypraios et al., 2017). Unlike point estimation methods that yield one estimate for each parameter in a model such as least squares, the aim of ABC methods is to estimate a distribution of parameter values that achieve some prespecified level of accuracy for a range of features. These target features must be defined by the investigator and could include quantities like the viral load peak, the time to viral load peak, time to viral clearance, or viral load set point (Swan et al., 2022).

Mathematically, the approach proceeds by simulating state variables $\mathit{X}(t)$ with the model using a proposed parameter list, ${\theta }^{*}$, where each value ${\theta }_k$ is drawn from a “prior” distribution, $p_k\left({\theta }_k\right)$; note that the distribution shape and values can be different for each parameter. The prior distribution (often called simply the “prior”) is assigned based upon prior knowledge about the process which may range from an educated guess to detailed empirical evidence. Then, a distance measure, $\rho$, (which can be some objective function, like RSS) is defined to compare model features to data features, for instance the peak of the model to the peak of the data. If the distance measure of all features falls within a prespecified tolerance value $(\varphi )$, then that parameter list is accepted (Fig. 3C). If repeated enough times, the posterior distribution of parameter values converges to an estimate of the population parameter distribution (Sunnåker et al., 2013; Turner and Van Zandt, 2012).

4.2. Algorithms and software

To retain realistic assumptions, viral dynamics models are usually and necessarily too complex to admit closed-form solutions. To optimize a model fit to observed data, the model must be repeatedly solved numerically, while updating parameter values to converge upon a best fit set as specified by the investigator. There are excellent existing computational algorithms to achieve model fitting which often provide guidelines to define convergence. Here we review three main choices.

4.2.1. Gradient descent

A natural approach to minimization of an objective function is to calculate the derivative of the objective function with respect to each parameter, and then move in the direction that achieves the largest decrease in the objective function. So-called gradient descent algorithms can be implemented in several ways to enhance speed and accuracy: Gauss-Newton and Levenberg-Marquardt are common methods (Lourakis, 2005; Gavin, 2019). The Gauss-Newton method tends to converge faster to the optimal solution compared to the Levenberg-Marquardt method, but it may fail to converge if the initial parameters are far from the optimum. Other methods allow a user to specify boundaries of parameter values, which can improve results and is particularly useful for enforcing biological feasibility of parameter estimates. Freely available implementations exist for many of these tools in common scripting languages like R and Python (SciPy: https://scipy.org/).

4.2.2. Markov Chain Monte Carlo (MCMC)

Markov Chain Monte Carlo (MCMC) simulation is a family of algorithms that sequentially samples from probability distributions until an equilibrium (or posterior) distribution is reached. The idea is to construct a sequence, or a “chain”, of random samples where each new sample depends on the previous one. (The name Markov indicates a process where the probability of each event depends only on the state of the previous event (Brooks et al., 2011; Gilks et al., 1995).) For parameter estimation, we specify an initial (prior) distribution for a parameter and then sequentially evaluate the model fit until a posterior distribution of acceptable parameters is found. The key to these methods is that the true distribution does not need to be known, but as the chain proceeds, the relative probabilities allow us to converge to a higher probability distribution.

As an example, a common MCMC approach is the Metropolis-Hastings algorithm. This process begins with a proposed model $\mathit{M}[\mathit{X}(t),\mathit{\theta },t]$ and initial parameter set ${\theta }^{(1)}$ for some individual $i$. Then, a new random set of parameters, ${\theta }^{(2)}$, is generated. The posterior probabilities of the new parameter sets, $P^{(1)}$ and $P^{(2)}$, are calculated using the likelihood of the parameter set (given the model) and Bayes’ theorem,

$$P(\mathit{\theta }\mid \mathit{D})\propto L(\mathit{\theta }\mid \mathit{D})P(\mathit{\theta })$$ (Eq. 10)

where $P(\mathit{\theta })$ is the prior probability of the proposed parameter set and $L(\mathit{\theta }\mid \mathit{D})$ is the likelihood of the data given the model with that parameter set. If the likelihood of the second parameter set is higher, i.e., $P^{(2)}>P^{(1)}$, then the proposed parameter set ${\theta }^{(2)}$ is naturally accepted. However, if the second set has a lower probability, rather than reject it, we accept it with probability $p=P^{(2)}/P^{(1)}$. This allows the algorithm to sometimes accept a slightly worse set of parameters such that later in the chain it can find a much better set (i.e., move out of a local minimum). The beauty of the algorithm arises because the proportionality (rather than equality) in Equation (10) is circumvented by taking the ratio of the probabilities.

The algorithm is repeated many times to generate a chain of samples that represents the possible values of the parameters of interest corresponding to maximum posterior probability. These samples can then be used to calculate point estimates, confidence intervals, and other important summaries about the parameters. Notably, the MCMC approach can provide a more comprehensive assessment of parameter uncertainty compared to other techniques, as it generates samples from the full posterior distribution of the parameters. For instance, de Pillis et al. used the MCMC algorithm to estimate the parameters in the within-host model of SARS-CoV-2 antibody dynamics after vaccination, leveraging longitudinal antibody titer data (de Pillis et al., 2023). That said, the MCMC approach can be computationally intensive, particularly for complex epidemiological and biological models that may consist of large unknown parameter sets to be estimated (Hamra et al., 2013).

4.2.3. Stochastic approximation expectation-maximization (SAEM)

The Stochastic Approximation Expectation-Maximization (SAEM) algorithm is a powerful technique for parameter estimation in pNLME models, especially for systems with missing or censored data ($D_{\text{cens}}$). The SAEM algorithm is based on the Expectation Maximization (EM) algorithm, which is an iterative procedure that alternates between an expectation (E) step, where the expected value of the log-likelihood of the complete data for an individual $i$ is $Q\left({\mathit{\theta }}^{(2)}\mid {\mathit{\theta }}^{(1)}\right)=E\left(L_{\text{cens}}\left(D,D_{\text{cens}};{\mathit{\theta }}^{(2)}\right)\mid D;{\mathit{\theta }}^{(1)}\right)$, and a maximization (M) step, where the model parameters are updated to maximize $Q\left({\mathit{\theta }}^{(2)}\mid {\mathit{\theta }}^{(1)}\right)$. Since the E step for a nonlinear model often does not have an analytical form, the stochastic version of the EM algorithm (SAEM) was introduced by Delyon et al. (1999). In SAEM, the E step includes the simulation of missing data and stochastic approximation (SA) of the likelihood of the complete data. The SA step gradually refines the estimates as more simulations are performed. This procedure repeats such that the algorithm converges to parameter values that best explain the experimental data.

Compared to the MCMC method, SAEM has been shown to be a very powerful and computationally efficient tool, especially for population models with many parameters, since it does not require sampling from the entire posterior distribution of parameters (Chan et al., 2011; Kuhn and Lavielle, 2005; Makowski and Lavielle, 2006). The SAEM algorithm is the backbone of the Monolix software suite and can also be used in the R package saemix.

4.2.4. Software

One may write their own code for model fitting and parameter estimation, but there are also many existing software packages available. Five options that are commonly used are given in Box 2.

Box 2. Software packages available for parameter estimation.

Monolix.

Monolix (https://lixoft.com/) is a software suite that is currently freely available for academic researchers. It performs pNLME with the SAEM algorithm and contains powerful diagnostic and visualization tools, such as point-and-click functionality to define which parameters should be estimated, choose covariates, specify what kind of noise exists in the data, and interactively visualize results.

NONMEM.

NONMEM (Nonlinear Mixed-Effect Modeling, https://www.iconplc.com/) is a long-standing software traditionally used for population pharmacokinetic (PK) and pharmacodynamic (PD) modeling and analyzing complex clinical and preclinical data. Though expensive, it is a field standard, especially in drug development and regulatory submissions. NONMEM supports a range of model structures, model diagnostics, and advanced estimation algorithms, such as the first-order conditional estimation and importance sampling to assess performance and guide refinement. It also includes simulation tools to predict drug concentrations and effects.

STAN.

STAN (https://mc-stan.org/) is an open-source probabilistic programming language for Bayesian inference and statistical modeling. It allows users to specify complex hierarchical models using intuitive syntax, and it efficiently performs sampling with advanced Markov Chain Monte Carlo (MCMC) methods. STAN supports a wide range of applications, from machine learning to neuroimaging, and handles various model types, including regression models. It integrates with languages such as R, Python, and MATLAB, providing user-friendly interfaces (e.g., rstan, pystan) and diagnostic tools to assess model convergence and fit. With its ability to manage complex models and large datasets efficiently, STAN offers both depth and flexibility, making it a leading tool for modern statistical and quantitative research.

DSAIRM.

Dynamical Systems Approach to Immune Response Modeling (DSAIRM, https://shiny.ovpr.uga.edu/DSAIRM/) is an R Shiny application developed by Handel (2020) specifically to reduce the barrier in mathematical modeling for immunologists working with simulation models. It provides a hands-on introduction to simulation models with examples of basic viral dynamics models.

MATLAB.

MATLAB (www.mathworks.com) is a powerful programming and computing platform with comprehensive capabilities for modeling, data analysis, simulation, and model fitting by all three approaches described in this review. Useful functions for model fitting include the iterative solvers fminsearch, fmincon, nlinfit, and nlmefitsa. The curve fitting toolbox (cftool) is also an excellent resource for exploratory model fitting.

• fminsearch: explores and converges towards a local minimum of a specified objective function.

• fmincon: finds the local minimum of a constrained objective function.

• nlinfit: fits nonlinear regression models using least squares estimation.

• nlmefitsa: fits nonlinear mixed-effects regression models using the stochastic EM algorithm.

5. Discussion and future directions

5.1. Conclusions: The optimal fitting strategy depends on data and scientific questions

Viral dynamic models are at their most powerful when testing scientific questions with competing models verified against appropriate data. However, data limitations are inevitable. Generally, model fitting is enhanced by raising the sampling frequency and increasing both the number of longitudinal timepoints and the number of individuals. Robust documentation of the sources of noise in data, and whether they differentially apply to low or high measured values, can be valuable.

Based on the present review, we propose a rough framework for choosing which model fitting approach is likely ideal (Fig. 5). Individual fitting has the advantages of a straightforward methodology, computational simplicity, and immediate transparency of individual model estimates. If there are few individuals but temporal sampling is dense, individual fitting can be highly successful. However, difficulties may arise when individuals are sampled at different frequencies, or when models are flexible enough such that parameter values are highly divergent across individuals. In cases of more variably sampled data across larger numbers of individuals, and/or when there are multiple study groups that can be differentiated by important covariates, it is often advantageous to use a population approach such that a coherent population average can be defined. Feature fitting is a relevant approach to handle more complex objective functions relating to many different data types, when models are necessarily stochastic, and when there are not many densely sampled individuals. There is no one-size-fits-all solution, and we believe that model fitting always requires thoughtful consideration of the scientific questions that the model is trying to answer and the types and amount of data at hand.

Fig. 5.

The quantity of available data can dictate which model-fitting approaches are applicable. When one only has aggregated data from multiple individuals or a very small number of data points per individual, then feature-based fitting provides a way forward for estimating parameter values. When one has a relatively small number of individuals (e.g., N < 10) and a substantial number of data points per individual (e.g., T ≥ 6 for the canonical viral dynamics model (Miao et al., 2011)), then fitting to each individual independently will likely be the best approach. When one has a large number of individuals with varying amounts of data available per individual, then pNLME will often be most suitable. There are also scenarios in which multiple approaches may work. Considerations regarding the best choice depend on the scientific question one is aiming to address with the model and/or the nature of the data itself. However, in the case of very limited data (e.g., few individuals and few time points per individual), estimating parameters exclusively from that data will likely fail. In these cases, further literature review and additional data are needed, or the model may need to be modified.

5.2. Future perspectives: Some further open questions and challenges include the following

5.2.1. Training vs. testing

In machine learning and statistics, much of the model fitting work we describe here would be referred to as training (Vasilevsky et al., 2013). Though often data-limited, we hope that future viral dynamics modeling can move towards cross-validation, where data not used to train the original model is used to test reproducibility and translatability. This will be more likely to be possible if modelers are directly involved in developing clinical protocols that generate relevant data.

5.2.2. Local vs. global minima

A frequent issue in nonlinear model fitting is that the objective function is often a complex, non-convex function with multiple local minima. This might lead the RSS to converge to a local minimum rather than the global minimum of the objective function, which could potentially lead to biased parameter estimates (Fig. 4B). Importantly, gradient descent algorithms naturally converge on the best parameter set near where it was initiated, rather than the absolute minimum across the entire parameter space. Though slightly more robust to this challenge (due to its stochastic algorithm), SAEM can be repeatedly run with different sets of initial conditions. Ultimately, it is never certain that a global minimum is achieved, but repeated fitting with multiple sets of initial conditions is recommended to mitigate returning to local minima.

5.2.3. Identifiability and fixing parameters based on prior literature

A common challenge for the individual fitting approach is that multiple (non-unique) solutions may arise. Parameter identifiability analysis aims to determine if unique parameter values can actually be inferred from data. Structural identifiability addresses this question in theory: If two parameters are completely correlated, then even perfect data will not allow their simultaneous estimation (Wieland et al., 2021). Moreover, even if the model is structurally identifiable (and they often are not), viral dynamics data may be sparse or limited by noise and thus admit practical identifiability issues. One pathway to identifiability that has been leveraged in many prior works is fitting viral dynamics models to data in which certain terms are effectively removed (i.e., when there is a forcing function (Conrad and Eisenberg, 2024)), the most common example being when antiviral therapy blocks viral infection, simplifying the equations and revealing clearance rates (Perelson et al., 1996; Martinelli, 2024). “Observability”, a concept from control theory, can also give insight into what parts of systems are safely estimated. From a sensitivity analysis perspective, certain estimated parameter values can sometimes vary over orders of magnitude without substantially influencing the goodness of fit, rendering the estimates mostly unreliable from a quantitative point of view, but also less relevant to model derived conclusions. Knowledge of accurate priors can forestall identifiability issues, but this may not always be possible. Many reviews have addressed overfitting and parameter identifiability concerns (Wu et al., 2008; Miao et al., 2011; Vasilevsky et al., 2013; Zitzmann et al., 2024). Though not completely universal, a quick method can be used to test identifiability in many cases relevant for viral dynamics (Castro and De Boer, 2020).

In viral dynamics, a common way to reduce the number of free parameters and overcome a lack of identifiability is to set (or “fix”) certain parameter values using prior information. However, prior parameter estimates rarely map directly, allowing the possibility of parameter value misclassification. It is critical to thoroughly review literature and consult with experts to determine values. In particular, if model parameters were originally inferred via a mathematical model, applying them to a differently structured model requires substantial care. It is often safer to define plausible ranges for fixed parameters and assess whether model conclusions are robust to varying through these ranges.

5.2.4. Iteratively building upon earlier models

Because modeling is iterative, newer and more sophisticated datasets may reveal a different model to be optimal for the same biological system. Thus, efforts to integrate information across continually updating models are needed to develop coherence in the field. A Bayesian approach (Gelman et al., 1995) whereby published model parameters are assumed to be posterior distributions and then used as prior distributions for future models could help incorporate previous knowledge (Yang and Rannala, 2010). In that vein, we advocate that published model fitting work should include full information on estimated posterior parameter distributions (e.g., distribution type, mean, variance, etc.).

Moreover, since it is not often possible to estimate all model parameters simultaneously from a single dataset, certain values must be imported from prior studies. Fixing certain values increases the identifiability of other parameter estimates. However, choosing a single value across all participants is likely an oversimplification. We caution that it has been historically common for modelers to insert parameters estimated via other modeling exercises, which could lead to propagation of incorrect and/or misclassified parameters when models change but prior values are used.

5.2.5. Model selection

A vital feature of modern model development is selection of multiple models for comparative testing, each encompassing a specific set of hypotheses. At present, this process is largely subjective and reflects traditional comparative hypotheses testing in experimental settings. A continual challenge is selecting the number of mathematical model variants. To comprehensively test all combinations, varying any equation terms means a multiplicative increase in the number of models considered. Testing thousands of competing models, each with a varying number of free parameters, can be slow and computationally expensive, and raises issues of reproducibility. To choose the optimal model, the standard process is to minimize information theoretic measures (Burnham and Anderson, 2004b) (e.g., Akaike and Bayesian Information Criteria, or AIC and BIC). These measures reward for fit while penalizing for complexity. Several models, however, often generate similar AIC and BIC scores (Byrne and Schiffer, 2024). Biological plausibility still remains critical for acceptable models, and model averaging or ensemble modeling provides another exciting possibility to integrate results from several good models (Byrne and Schiffer, 2024; Gonçalves et al., 2020). In the future, it may be possible to leverage language models to scan literature for equation terms and thereby determine large fields of candidate models. In the case of excellent data measured from several state variables, it may also be possible to apply sparse matrix inversion techniques to agnostically infer model terms directly from data (Somacal et al., 2022; Fasel et al., 2022; Brunton et al., 2016).

5.2.6. Spatially structured models

There are many aspects of viral infection dynamics that are inherently spatial, and therefore, not accurately captured by mean-field ODE models. These aspects include some immune system function, cell-to-cell transmission, virus diffusion, and spread of infection between different tissues (Williams et al., 2025). Spatially structured models, including partial differential equations (PDEs) (Bocharov et al., 2016; Quirouette et al., 2020), Agent-Based Models (ABM) (Roychoudhury et al., 2020), and patch models (Schiffer et al., 2013a, 2013b, 2016a, 2018; Schiffer, 2013), have been used to include some of these aspects. In many cases these models exhibit different dynamics compared to their ODE counterparts in the same parameter regime, highlighting the limitation of ODE models in those scenarios (Strain et al., 2002; Funk et al., 2005; Mitchell et al., 2011; Marzban et al., 2021).

The inherent complexity of spatially structured models, difficulty in obtaining spatial data, and challenges in conducting rigorous methods of parameter estimation limit the use of these models (Williams et al., 2025). Sometimes, spatial model parameter values are taken from estimates via prior ODE models. However, the change in exactly what each parameter means in the spatial setting can make this approach unreliable. Feature fitting to spatial data summary measures is of particular relevance for spatial models. Latin hypercube or grid searches through parameter sets coupled with fit (RSS) to viral load summary measures has been successful to estimate multiple parameters in the HSV-2 context (Roychoudhury et al., 2020).

5.2.7. Model fitting to multicomponent viral load and immune response data

The availability of new data types represents a particularly exciting frontier for viral dynamics. In the past, multiple model equations were generated to reproduce one state variable (typically viral load), while predicted trajectories of other state variables were rarely tested. At best, models were fit to viral load and one measure of immune response (Yang et al., 2021; Clairon et al., 2023; Tessandier et al., 2025). Modern research protocols now may include serial assessment of dozens of cytokines (Lim et al., 2023, 2025; Waghmare et al., 2022), thousands of RNA transcripts relating to cellular immune responses, measures of multiple pathogen-specific antibody and T cell responses (Vick et al., 2021), and values of levels of small molecules and immunotherapeutics (Goyal et al., 2022b; Esmaeili et al., 2025). While these data present extraordinary opportunities for model validation, as well as complex data fitting, a novel problem is that the number of experimental variables may far exceed the number of state variables in the models, rather than vice versa. Future efforts will need to focus on data reduction techniques and updated data fitting strategies to allow for recapitulation of detailed multi-component immune response data, to match contemporaneous shifts in observed viral load.

5.2.8. Model forecasting and prediction

Viral dynamics models that have been calibrated by comparing model output to experimental data can be used to identify real-world rates or interactions that are not intuitive from statistical analysis of raw nonlinear data. A major advantage of viral dynamics models is that they can often be trained against longitudinal measurements from multiple individuals. Thus, a well-validated model represents individual trajectories of viral loads accurately, but also encompasses inter-individual heterogeneity. In our opinion, strong models can be used to help understand deeper connections in observed data, generate future hypotheses for testing, and interpolate the behavior of unobserved state variables (with caution).

In the absence of data for validation, models should generally be considered less reliable for extrapolation beyond measured data. However, this is dependent on context. Within an individual, it has yet to be shown that the course of infection can be predicted based on prior data (Dhankani et al., 2014). However, a well validated model of treatment with a given dose may accurately predict trial outcomes with other untested doses (Schiffer et al., 2016b). Overall, the ability of a model to forecast outcomes should be validated with relevant data.

6. Summary

We reviewed multiple approaches to fitting models of viral dynamics and parameter estimation. Furthermore, we highlighted ongoing changes in the field, which will require refinement of these strategies.