Impact of reproduction number on the multiwave spreading dynamics of COVID-19 with temporary immunity: A mathematical model

Abstract

Objectives: The recent discoveries of phylogenetically confirmed COVID-19 reinfection cases worldwide, together with studies suggesting that antibody titres decrease over time, raise the question of what course the epidemic trajectories may take if immunity were really to be temporary in a significant fraction of the population. The objective of this study is to obtain an answer for this important question.

Methods: We construct a ground-up delay differential equation model tailored to incorporate different types of immune response. We considered two immune responses: (a) short-lived immunity of all types, and (b) short-lived sterilizing immunity with durable severity-reducing immunity.

Results: Multiple wave solutions to the model are manifest for intermediate values of the reproduction number R; interestingly, for sufficiently low as well as sufficiently high R, we find conventional single-wave solutions despite temporary immunity.

Conclusions: The versatility of our model, and its very modest demands on computational resources, ensure that a set of disease trajectories can be computed virtually on the same day that a new and relevant immune response study is released. Our work can also be used to analyse the disease dynamics after a vaccine is certified for use and information regarding its immune response becomes available.

Keywords: COVID-19 reinfection; Multiple wave solutions; SEIRS-type delay model; Simple and complex immune response.

Graphical abstract

Figure 1

City A eliminated the epidemic via non-pharmaceutical interventions alone.

Figure 2

City C reached herd immunity well before the 200-day mark. ‘k’ Denotes thousand and ‘L’ denotes hundred thousand.

Figure 3

City B1 (we used B2 to denote the same city with a complex immune response) was a less effective form of A and had multiple waves of COVID-19. ‘k’ Denotes thousand and ‘L’ denotes hundred thousand. We stopped the simulation at 1400 days but it is important to note that the waves persisted after this time.

Figure 4

City F took a bold approach and “beat the virus to the finish line”. ‘k’ Denotes thousand and ‘L’ denotes hundred thousand.

Figure 5

City G1 represents a failed version of City F; the error in this case carries a significant price. ‘k’ Denotes thousand and ‘L’denotes hundred thousand.

Figure 6

City B with a complex immune response. ‘k’ Denotes thousand and ‘L’ denotes hundred thousand. W/7 denotes the weekly increments in cases scaled down by 7 as described previously, while HVF and LVF refer to the higher and lower virulence forms of the disease, respectively. y Is shown in blue, $\dot{y}$ in green, z in red, $\dot{z}$ in magenta, and the epi-curves in grey for y and cyan for z.

Figure 7

City G with a complex immune response. In the right third of the plot, the cyan bars obscure the smaller grey bars; the latter are enveloped by the green curve which is still visible. ‘k’ Denotes thousand and ‘L’ denotes hundred thousand. W/7 denotes the weekly increments in cases scaled down by 7 as described previously, while HVF and LVF refer to the higher and lower virulence forms of the disease, respectively.