Zoonotic pathogens represent a growing global risk, yet the speed of adaptive immune activation across mammalian species remains poorly understood. Despite orders-of-magnitude differences in size and metabolic rate, we show that the time to initiate adaptive immunity is remarkably consistent across species. To understand this invariance, we analyse empirical data showing how the numbers and sizes of lymph nodes scale with body mass, finding that larger animals have both more and larger lymph nodes. Using scaling theory and our mathematical model, we show that larger lymph nodes enable faster search times, conferring an advantage to larger animals that otherwise face slower biological times. This enables mammals to maintain, or even accelerate, the time to initiate the adaptive immune response as body size increases. We validate our analysis in simulations and compare it to empirical data.

To validate our mathematical model, we implement an agent-based model (ABM), termed the Initial First Contact Time (IFCT) model, using the MASON libraries [1] in Java. The model simulates two types of agents - T cells (searchers) and Dendritic Cells (DCs) (targets). Both are uniformly distributed within a cubic representation of a Lymph Node (LN). LN space is modeled as a continuous Cartesian grid in three dimensions with fixed reflective boundaries.

The IFCT model is run as a discrete-time simulation, with each time step representing one second. At the beginning of each simulation run, a predefined number of T cells and DCs are initialized (see \Cref{tab:inputDataTable}). The positions of DC are static throughout the simulation. T cells, on the other hand, move in each time step with either Brownian motion or a persistent random walk, which was modeled from empirical data in Fricke et al.[2]. We model T cell motion without considering collisions so that T cells pass through each other.

We run the IFCT model for the two bounding cases, one in which N_DC is proportional to M⁰, i.e., a constant, and another where N_DC  is proportional to M^0.5. Each case is simulated for five different animal mass (M) values. For each M, we run 100 experimental replicas for each combination of factors, where the initial distribution of the T cells and DC is stochastic. We assume that there is a contact between a T cell and DC when their centers are within 10 microns. The simulation result from the IFCT model confirms that the distribution of contact times is exponential, an assumption that underpins the theoretical derivation.

[1] S. Luke, G. C. Balan, L. Panait, C. Cioffi-Revilla, S. Paus, MASON: A Java multi-agent simulation library, in Proceedings of Agent 2003 Conference on Challenges in Social Simulation, vol. 9 (2003).

[2] G. M. Fricke, K. A. Letendre, M. E. Moses, J. L. Cannon, Persistence and adaptation in immunity: T cells balance the extent and thoroughness of search. PLoS Computational Biology 12 (3), e1004818 (2016).