1. It can be difficult to define what something is, so instead: what is something that looks like mathematical immunology, but is not?

• A.S.: Mathematical immunology by name suggests that it’s a model of the host response. We can likely go one step further and define it as a model focused on answering immunological questions. I would not consider any statistical model or other non-mechanistic approach as mathematical immunology.

• D.R.: Large caveat: I don’t want to disparage any research and I’m finding it hard to write this exclusionary definition without it sounding dismissive. Two ideas: (1) Large computational clustering exercises. Though UMAP relies on beautiful topology, or mathematics, the way it is used to cluster transcriptomic data to define types of cells is not often mechanistically revelatory, which I think is a key for good mathematical immunology. (2) Deriving R0s. This is a controversial one. But I’ll posit that the beauty and insight of mathematical immunology arises from the complexity and dynamics of the models we examine. Thus, analyzing very early dynamics (especially in models in which an immune population will develop eventually) rarely provides the right level of biological insight we need to communicate with experimentalists. Sort of like in physics, *non-equilibrium* statistical mechanics is often required to understand real systems. Implicit in this opinion is that without fostering interdisciplinary connections (and therefore learning to communicate), we can’t do mathematical immunology well.

• F.A.: A purely machine-learning analysis of a large data set, such as a time series of cytokines and gene expression, that identifies patterns without an underlying mechanistic model.

• M.C.: Models that incorporate “immune cells” in the most general sense (e.g., to remove infected cells) but without reference to cellular types or function to complexify a classic/existing model. Essentially, these are just predatory prey models (as are all of our CD8+ T cell models) but many times, this approach is used to add complexity for performing the type of mathematical analysis that interests the authors and not to forward our understanding of immunology. There is room for both approaches in mathematical biology, but I think that at its core, mathematical immunology should help us understand immunology.

• S.C.: I do not consider statistical and phenomenological approaches that merely quantify data and identify correlates without proposing mechanistic hypotheses, to be part of mathematical immunology. Furthermore, large-scale computational models that incorporate numerous parameters—for cytokines, chemokines, and signaling pathways—are more appropriately classified within the realm of computational immunology.

2. List 1–2 key papers or models that, in your view, define mathematical immunology.

• A.S.: Alan (Perelson)’s early papers describing antibody shape space [10.1016/0165-2478(89)90175-2, 10.1016/B978-0-12-287960-9.50027-7]. This work was ahead of its time and was validated decades later. Perhaps also the collective work of those studying T cell fate - Rob De Boer has a nice recent review about the advances in modeling with a relatively unifying model infrastructure [10.1146/annurev-immunol-101721-040924].

• D.R.: This is way too hard to pick. These were all influential on me and range from branching process, stochastic differential equations, complex ODEs, model fitting, immunology/infectious disease/therapies. (1) Hill et al. [10.1073/pnas.1406663111]. This one is a nice example of a simple dynamical system that applies a stochastic model and tackles a hard but critically important question about defining HIV cure. It features some nice analytical calculations that end up using a Pochhammer symbol (rising factorial!), stochastic simulations, and a strong collaboration. It connects back with the few known cases of HIV cure at the time, a nice way to draw attention to the reality of data limitations but also showing the model projects (plausibly) data not used to train it. (2) Desponds et al. [10.1073/pnas.1512977112]. This is another elegant/simple modeling paper that addresses a question I’ve been fascinated by, or perhaps even was the genesis of this fascination, which is what rules govern the diversity/abundances of T cell clonotypes in the immune system. It uses stochastic differential equations (I am very partial because I worked with these a lot during my PhD) and posits an interesting idea of a “fluctuating fitness landscape”, i.e. a variable antigenic stimulus that allows some clones to proliferate more than others at different times. It does some nice mathematical work to derive that a power law relationship in clone sizes emerges from a Fokker-Planck equation but that a slightly more complicated model is needed. This and other models like it have impacts across many theoretical immunology studies since then. (3) Desikan et al. [10.1371/journal.pcbi.1008064]. I think this is a fantastic example of attempting to match a complex theory with real data. The model is based on Conway and Perelson 2015, (which I could also have picked!), but I wanted to highlight that it’s a great feature of math biology how a model can build on another. Here they take that tricky bistable model, which inherently is extremely sensitive to parameter values, and manage to fit impressively well the oscillating trajectories from a macaque study of post-treatment control using broadly neutralizing antibodies as a perturbation. It features some nice identifiability analysis and an early (for me) example of usage of Monolix, which has of course become quite prevalent in recent years.

• F.A.: One of the Perelson papers on influenza dynamics, such as Baccam et al. [10.1128/jvi.01623-05]. This paper is one of the first to build some simple immunological dynamics on top of standard viral dynamics, and to validate the predictions with data. This inspired many of us to realize that one can usefully include immunology without being overwhelmed by complexity.

• M.C.: (1) Perelson, Kirschner, De Boer (10.1016/0025-5564(93)90043-A). (2) Myers et al. (10.7554/eLife.68864)

• S.C.: I have attached a 2025 paper that perfectly describes modeling of vaccine induced immunity. I am also attaching two historical papers that shaped my thinking about the modeling of humoral and cellular immunity. (1) [10.3389/fimmu.2025.1596518] The Ribeiro paper (based roughly on the historical Oprea paper [10.1006/bulm.1999.0144]) has several important flavors: a detailed model of B cell and plasma cell dynamics (that can offer mechanistic predictions), a great data set that asks questions regarding antibody responses following vaccination (across pathogens), and the novel use of machine learning for forecasting and to tease out differences between responses (in this case was antigen presentation). (2) [10.1128/jvi.75.22.10663-10669.2001] The T cell paper from de Boer’s group, was (one of the) first ones that looked at CD8 T cell dynamics (they later did CD4 and CD8) following pathogen resolution, and modeled the expansion and contraction of activated cells and the generation of memory. It was especially interesting that they matched it with LCMV data.

3. If given sufficient time and resources, how would you improve a classical result using modern capabilities?

It’s not surprising that there is no clear consensus here. This really highlights the near-infinite possibilities for mathematical modelers who are willing to dive deeper into modern immunology.

• A.S.: I’m fairly convinced that most of science today is repeating/expanding on what others did 25-100+ years ago with better resources (but less time & money!). Mathematically, the viral dynamics model could be argued as a prime example. It’s widely used for obvious reasons but has very little power regarding viral-host interactions. Developing models of immunity is difficult and having the right amount in addition to the right type of data to be discriminatory and predictive is extremely challenging and expensive. Another, possibly more controversial one, is the idea behind cellular stochasticity. The single-cell techniques will likely show that this is not true stochasticity but rather tightly regulated and predictable switches. Immunologically, I think revisiting complement activation and concepts of cell exhaustion could benefit from newer techniques.

• D.R.: I would gather data from the early 90s on how AIDS develops and come up with a much more comprehensive and data-driven model of the immunological/evolutionary mechanisms leading to progression from HIV to AIDS.

• F.A.: Use a variety of omics approaches to build a larger patient cohort to link intercellular dynamics with mechanisms within the cell. It would be particularly useful to understand how prior exposure to related infections is encoded in the immune system and how that shapes the subsequent response.

• M.C.: Integrating data that leverages improved experimental techniques (e.g., better sub-setting of cells, genetic classifications etc.) combined with newer techniques for parameter estimation and sensitivity analyses would allow for an improved understanding of how biological heterogeneity impacts results. I think we need to be moving away from “average” cases to understand the breadth of responses seen in experiments and in the clinic.

• S.C.: Some ideas: Organize a study aimed at reconciling results across different species (mice, non-human primates, and humans) and developing methods to appropriately scale these findings. The models would be expanded to incorporate data from key immune compartments such as lymph nodes, germinal centers, and the thymus; to integrate longitudinal data capturing the effects of aging on immune dynamics; to model the maintenance of immune memory over time; and to specifically address sex-based differences in immune responses.

About our interviewees:

Amber Smith (A.S.) is an associate professor in the Department of Pediatrics and the Institute for the study of Host-Pathogen Systems at the University of Tennessee Health Science Center.

Daniel Reeves (D.R.) is a principal staff scientist in the Infectious Disease Sciences Program Vaccine and Infectious Disease Division at Fred Hutchingson Cancer Center, and an affiliate assistant professor at University of Washington.

Fred Adler (F.A.) is a professor in Mathematics Department and the School of Biological Science at University of Utah.

Morgan Craig (M.C.) is a researcher at the Sainte-Justine University Hospital Azrieli Research Centre, and an Associate Professor in the Department of Mathematics and Statistics at the Université de Montréal.

Stanca Ciupe (S.C.) is a professor in the Department of Mathematics at Virginia Tech.