Stochastic Models and Estimation of Undetected Infections in the Transmission of Zika Virus

TL;DR

An enhanced SEIR (Susceptible-Exposed-Infectious-Recovered) model is provided to incorporate partial information by distinguishing between detected and undetected Zika infections (also known as"dark figures").

Abstract

Zika fever, a mosquito-borne viral disease with potential severe neurological complications and birth defects, remains a significant public health concern. The epidemiological models often oversimplify the dynamics of Zika transmission by assuming immediate detection of all infected cases. This study provides an enhanced SEIR (Susceptible-Exposed-Infectious-Recovered) model to incorporate partial information by distinguishing between detected and undetected Zika infections (also known as "dark figures"). By distinguishing the compartments, the model captures the complexities of disease spread by accounting for uncertainties about transmission and the number of undetected infections. This model implements the Kalman filter technique to estimate the hidden states from the observed states. Numerical simulations were performed to understand the dynamics of Zika transmission and real-world data was utilized for parameterization and validation of the model. The study aims to provide information on the impact of undetected Zika infections on disease spread within the population, which will contribute to evidence-based decision making in public health policy and practice.

Figures (18)

• Figure 1: Simplified Zika virus transmission dynamics considering gender (male human$( {m})$, female human$({f}$)) and vector (${v}$).The blue compartments indicate the observable states, while the red compartments are the unobserved states (dark figures). The green dashed lines indicate the vector-to-human transmission while red lines indicate the human-to-vector transmission and blue lines indicate the human-to-human transmission. A black solid arrow shows the transition from one compartment to the next.
• Figure 2: Base Zika model: virus transmission dynamics considering gender (male human$({m})$, female human $({f})$ and vector $({v})$.The blue compartments indicate the observable states for the males and females, while the red compartments are the unobserved states (dark figures). The green dashed lines indicate the vector-to-human transmission while red lines indicate the human-to-vector transmission and blue lines indicate the human-to-human transmission. The transition from one compartment to the next is shown by the black solid arrow.
• Figure 3: Extended Zika virus transmission model highlighting the inclusion of $L = 3$ cascaded recovery compartments $R^1_m \to R^2_m \to R^3_m$ for males and $R^1_f \to R^2_f \to R^3_f$ for females. All these cascade compartments are included in the observed states and other compartmental transitions remain as defined in the base model.
• Figure 4: Monthly rainfall pattern in Rio de Janeiro, Brazil, showing clear seasonality with peaks during the summer months (December–March) and lows in the winter (June–August)${}^{\ref{['fn_weather']}}$. The solid line represents the mean rainfall computed over a 31-day sliding window, while the shaded regions indicate empirical percentile bands: the darker band corresponds to the interquartile range (25th–75th percentiles), and the lighter band spans the 10th–90th percentiles.
• Figure 5: Calibration of the mosquito birth rate $B_v(t)$ from transformed rainfall data in Rio de Janeiro. The function reflects seasonal variability in mosquito reproduction driven by environmental conditions.

Theorems & Definitions (12)

• remark thmcounterremark
• theorem 4.1: Liptser & Shiryaev (2001) LiptserShiryaevVolII2001
• proof
• theorem 4.2: Liptser & Shiryaev (2001) LiptserShiryaevVolII2001
• proof
• remark thmcounterremark: Use of the Moore–Penrose Pseudoinverse
• remark thmcounterremark: Online Covariance Updates in Nonlinear Systems
• lemma thmcounterlemma
• proof
• remark thmcounterremark