An immuno-epidemiological model for transient immune protection: A case study for viral respiratory infections

Abstract

The dynamics of infectious disease in a population critically involves both within-host pathogen replication and between host pathogen transmission. While modeling efforts have recently explored how within-host dynamics contribute to shaping population transmission, fewer have explored how ongoing circulation of an epidemic infectious disease can impact within-host immunological dynamics. We present a simple, influenza-inspired model that explores the potential for re-exposure during a single, ongoing outbreak to shape individual immune response and epidemiological potential in non-trivial ways. We show how even a simplified system can exhibit complex ongoing dynamics and sensitive thresholds in behavior. We also find epidemiological stochasticity likely plays a critical role in reinfection or in the maintenance of individual immunological protection over time.

Keywords: Flow-kick dynamics; Immune boosting; Immuno-epidemiology; Priming number; Viral-immune mathematical model.

Fig. 1

The virus (V), target cells (T), and infected cells (I) constitute a three-variable subsystem wherein virus infects target cells to produce infected cells and infected cells produce more virus. The innate immune function is represented by interferon (F) which is produced in the presence of infected cells and inhibits viral production by killing infected cells. Adaptive immunity in the model consists of B-cell (B) activation in the presence of virus, antibody (A) production in the presence of B-cells, and neutralization of viri by antibody.

Fig. 2

Short term (10 day) immune response to a single viral exposure. (A)–(F) Baseline solution to the continuous system given in Equation (1) with an initial condition of V = 15000, T = C_T, and all other variables equal to 0.

Fig. 3

(A)–(F) Long term (400 day) immune response with a second viral exposure at τ = 283. (A)–(C) The viral dose is set to k = 14800, and there is no excursion after the second exposure. This corresponds a point in the blue region of (G). (D)–(F) The viral dose is set to k = 15000. This is the smallest (integer valued) second dose that triggers a second excursion at τ = 283. This corresponds a point outside of the blue region of (G). (G) The viral re-exposure dose needed to trigger a second excursion depends on the timing of re-exposure. The shaded region shows the kick sizes (doses) that do not trigger a second excursion for each re-exposure time τ (5 ≤ τ ≤ 400).

Fig. 4

Long term (400 day) immune response with viral re-exposure every 7 days. (A)–(F) Case 1: iteration of the flow-kick system with (τ, k) = (7, 15000). This combination of parameters (τ, k) leads to a stable flow-kick equilibrium corresponding to a protected state in which no reinfection occurs. For a finer time-scale depiction of this equilibrium, see Supplementary Figure A1 A-F. (G)–(L) Case 2: iteration of the flow-kick system with (τ, k) = (7, 16244), representing a state in which reinfection does occur. This is the smallest integer kick size that yields a second excursion when applied every 7 days. To see a finer time-scale depiction of the dynamics near the second excursion, see Supplementary Figure A1 G-L.

Fig. 5

The periodic viral re-exposure dose required to provide long-term protection against reinfection depends on the inter-exposure interval. The blue area corresponds to parameter values (τ, k) for which the deterministic flow-kick system has a stable flow-kick equilibrium, i.e. the combination of inter-exposure interval and dose magnitude provides long-term protection against reinfection. The small yellow box depicts the set 7 < τ < 8 and 12000 < k < 13000 from which the viral re-exposure dose and inter-exposure period are randomly sampled (uniformly) in the stochastic simulation in Fig. 6.

Fig. 6

Stochastic simulation of the flow-kick system. (A) An example of stochastic kicks and viral exposures sampled uniformly from the set 7 < τ < 8, 12000 < k < 13000 depicted in Fig. 5. (B)–(G) A subset of stochastic simulations are plotted here. Out of 500 simulations, 386 (77%) had an excursion representing reinfection by t = 600.

Fig. A1

(A)–(F) A flow-kick equilibrium corresponding to Fig. 4 A-F, with τ = 7, k = 15000. This is an example of protection against deterministically periodic repeated re-exposure. (G)–(L) The dynamics close to excursion in a flow-kick system with τ = 7, k = 16244. This corresponds to Fig. 4 G-L.

Fig. A2

This figure colors the complement of Fig. 5. For each (τ, k), the color indicates the iteration number of the deterministic flow-kick system associated with the first re-infection. The white region on the lower left is associated with locations that never show reinfection, corresponding to the blue protection region in Fig. 5. There is banding in the iteration number, starting at iteration number 17 on the upper right and going to iteration number 30 on the upper left. Regions associated with constant iteration numbers are separated by regions associated with canard-like behavior.