Epidemiological impacts of post-infection mortality

Abstract

Infectious diseases may cause some long-term damage to their host, leading to elevated mortality even after recovery. Mortality due to complications from so-called 'long COVID' is a stark illustration of this potential, but the impacts of such post-infection mortality (PIM) on epidemic dynamics are not known. Using an epidemiological model that incorporates PIM, we examine the importance of this effect. We find that in contrast to mortality during infection, PIM can induce epidemic cycling. The effect is due to interference between elevated mortality and reinfection through the previously infected susceptible pool. In particular, robust immunity (via decreased susceptibility to reinfection) reduces the likelihood of cycling; on the other hand, disease-induced mortality can interact with weak PIM to generate periodicity. In the absence of PIM, we prove that the unique endemic equilibrium is stable and therefore our key result is that PIM is an overlooked phenomenon that is likely to be destabilizing. Overall, given potentially widespread effects, our findings highlight the importance of characterizing heterogeneity in susceptibility (via both PIM and robustness of host immunity) for accurate epidemiological predictions. In particular, for diseases without robust immunity, such as SARS-CoV-2, PIM may underlie complex epidemiological dynamics especially in the context of seasonal forcing.

Keywords: epidemiological model; periodicity; post-infection mortality.

Figure 1.

Formulation of epidemiological model with PIM. (a) Schematic illustration of epidemiological processes that are encompassed in the model framework. (b) Model flow diagram with rates into/out of each compartment.

Figure 2.

Illustrative examples of time series of S_P(t), I(t) and S_S(t) for the periodic behaviour that can arise due to PIM, for different transmission rates β and different PIM rates α_S. Since γ = 1 and $\Lambda =\mu =\frac{1}{50(52)}$, ${\mathcal{R}}_0=\beta /(1+\mu )\approx \beta$. For (a,c,e) and (g), we simulate 600 years with the first week having I(t₀) = 10⁻⁹, S_P(t₀) = 1 − I(t₀) and S_S(t₀) = 0, and we plot weeks 400(52) + 1 to 600(52). Panels (b,d,f) and (h) present the values of the complex-conjugate pair of eigenvalues of the Jacobian matrix of the S_S, I and N equations about the endemic equilibrium, for different values of PIM α_S. The ‘star’ symbol in each of these plots corresponds to the value of α_S used for the corresponding previous plot. In all four panels, α_I = 0 and $\epsilon =1$.

Figure 3.

Illustrative examples of PIM triggering endogenous periodicity. Heat maps of time series of (a) S_P, (b) S_S and (c) I as PIM is varied. In all panels, β = 2.5, γ = 1, α_I = 0, $\mu =\frac{1}{50(52)}$ and $\epsilon =1$. To obtain these heat maps, we simulate 500 years with initial conditions (at the first week) of I(t₀) = 10⁻⁹, S_P(t₀) = 1 − I(t₀) and S_S(t₀) = 0. For the heat maps, we discard the first 400 years and plot weeks 400(52) + 1 to 500(52).

Figure 4.

Interplay of additional host and pathogen characteristics with PIM. (a–d) Illustrative example for the impact of host immunity on epidemic dynamics with PIM. The heat maps in (a–c) are as in figure 3, but with $\epsilon$ varying instead of α_S. (d) As in figure 2b,d,f,h, but with $\epsilon$ varying. In (a–d), β = 2.5, α_S = 0.001, γ = 1, α_I = 0 and $\mu =\frac{1}{50(52)}$. (e–h) Illustrative example for the impact of disease-induced mortality during active infection. Panels (e–g) are as in figure 3, but with α_I varying instead of α_S. (h) As in that of (d), but with α_I varying instead of $\epsilon$. In (e–h), β = 2.5, $\epsilon =1$, α_S = 0.00065, $\mu =\frac{1}{50(52)}$, γ = 1 and $\mu =\frac{1}{50(52)}$. For the heat maps of (a–c) and (e–g), and as in figure 3a, weeks 400(52) + 1 to 500(52) are plotted.