Modelling cholera in periodic environments

Abstract

We propose a deterministic compartmental model for cholera dynamics in periodic environments. The model incorporates seasonal variation into a general formulation for the incidence (or, force of infection) and the pathogen concentration. The basic reproduction number of the periodic model is derived, based on which a careful analysis is conducted on the epidemic and endemic dynamics of cholera. Several specific examples are presented to demonstrate this general model, and numerical simulation results are used to validate the analytical prediction.

Fig. 4.

A typical infection curve for each model when R ₀<1, with initial condition I(0)=1. The solution quickly converges to the disease-free equilibrium with I ₀=0. (a) Model 6.1, (b) Model 6.2, (c) Model 6.3, (d) Model 6.3 in long term.

Fig. 5.

A typical infection curve for each model when R ₀>1, with initial condition I(0)=1. A periodic solution with ω=365 days forms after a long transient in each case. (a) Model 6.1, (b) Model 6.2, (c) Model 6.3, (d) Model 6.2 zoom-in.

Fig. 1.

Plots of the periodic threshold of R ₀ for various ā and ã, respectively, in model 6.1. (a) R ₀=1 when ā ≈ 0.0625, and [R ₀]=1 when ā ≈ 0.0667; (b) R ₀=1 when ã ≈ 0.8407, and [R ₀]=0.90 for all ã.

Fig. 2.

Plots of the periodic threshold of R ₀ for various β_ē and β_e˜, respectively, in model 6.2. (a) R ₀=1 when β_ē ≈ 0.0321 and [R ₀]=1 when β_ē ≈ 0.0334; (b) R ₀=1 when β_e˜ ≈ 0.5688 and [R ₀]=0.9797 for all β_e˜.

Fig. 3.

Plot of the periodic threshold of R ₀ for various b _W˜ in model 6.3. R ₀=1 when b _W˜ ≈ 0.3706 and [R ₀]=0.9872 for all b _W˜.