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Age-structured model of dengue transmission dynamics with time-varying parameters, and its application to Brazil

Ihtisham Ul Haq, Serge Richard

TL;DR

This work develops a time-varying, age-structured host–vector model for dengue that explicitly tracks asymptomatic and symptomatic human infections, vaccination, and vector dynamics. It derives disease thresholds using renewal theory via Euler–Lotka and next-generation matrix approaches, linking the basic and time-varying reproduction numbers to the instantaneous growth rate $r_t$. The authors apply the framework to Brazil (2021–2024), estimating medical and transmission parameters from weekly data and climate indicators, and demonstrate forecasting capabilities under data-driven and climate-driven transmission rates. The study highlights the importance of age structure, vector ecology, and climate in shaping dengue dynamics and offers a platform for data-informed policy and outbreak forecasting, while acknowledging data limitations and model simplifications.

Abstract

An age structured mathematical model with time dependent parameters is developed to investigate the dynamics of dengue transmission. Its properties are thoroughly analyzed in the first part of this work, as for example its disease free steady state, the corresponding effective reproduction numbers, its basic reproduction number (obtained via the Euler and Lotka equation and the next generation matrix approach). We also provide formulas for the time-varying effective reproduction number, and draw relations with the instantaneous growth rate. In the second part, we apply this model to Brazil and use weekly time series data from this country. Various medical parameters are firstly evaluated from these data, and an extensive numerical simulations for the period 2021 to 2024 is then carried out. Estimation of the transmission rates are derived both from epidemiological data and from environmental data such as temperature and humidity. The time-varying effective reproduction numbers are then estimated on these data, following the theoretical investigations performed in the first part. The sensitive parameters that significantly affect the model dynamics are presented graphically. Model predictions for following year by using different transmission rates are finally presented. Our findings show the importance of population age distribution, vector population dynamics, and climate, contributing to a deeper understanding of dengue transmission dynamics in Brazil.

Age-structured model of dengue transmission dynamics with time-varying parameters, and its application to Brazil

TL;DR

This work develops a time-varying, age-structured host–vector model for dengue that explicitly tracks asymptomatic and symptomatic human infections, vaccination, and vector dynamics. It derives disease thresholds using renewal theory via Euler–Lotka and next-generation matrix approaches, linking the basic and time-varying reproduction numbers to the instantaneous growth rate . The authors apply the framework to Brazil (2021–2024), estimating medical and transmission parameters from weekly data and climate indicators, and demonstrate forecasting capabilities under data-driven and climate-driven transmission rates. The study highlights the importance of age structure, vector ecology, and climate in shaping dengue dynamics and offers a platform for data-informed policy and outbreak forecasting, while acknowledging data limitations and model simplifications.

Abstract

An age structured mathematical model with time dependent parameters is developed to investigate the dynamics of dengue transmission. Its properties are thoroughly analyzed in the first part of this work, as for example its disease free steady state, the corresponding effective reproduction numbers, its basic reproduction number (obtained via the Euler and Lotka equation and the next generation matrix approach). We also provide formulas for the time-varying effective reproduction number, and draw relations with the instantaneous growth rate. In the second part, we apply this model to Brazil and use weekly time series data from this country. Various medical parameters are firstly evaluated from these data, and an extensive numerical simulations for the period 2021 to 2024 is then carried out. Estimation of the transmission rates are derived both from epidemiological data and from environmental data such as temperature and humidity. The time-varying effective reproduction numbers are then estimated on these data, following the theoretical investigations performed in the first part. The sensitive parameters that significantly affect the model dynamics are presented graphically. Model predictions for following year by using different transmission rates are finally presented. Our findings show the importance of population age distribution, vector population dynamics, and climate, contributing to a deeper understanding of dengue transmission dynamics in Brazil.

Paper Structure

This paper contains 33 sections, 4 theorems, 163 equations, 28 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Theoretical investigations
  3. The model
  4. The rescaled model
  5. Disease-free steady state
  6. Basic effective reproduction number and basic reproduction number
  7. First approach for estimating the basic effective reproduction number
  8. Second approach for estimating the basic effective reproduction number
  9. Third approach for estimating the basic effective reproduction number
  10. Interpretation of the expression obtained
  11. Comparison of basic reproduction number obtained with the Lotka equation and with the next generation matrix approach
  12. Derivation of the basic reproduction number using the Euler-Lotka approach
  13. Derivation of the basic reproduction number using the next generation matrix approach
  14. Stability analysis
  15. Time-varying effective reproduction number
  16. ...and 18 more sections

Key Result

Lemma 4.1

Consider the system section1.13 with $\mu_{H}=0$ and with initial conditions at the DFE, namely and boundary conditions Then for all $(t,\zeta)\in [t^0,\infty)\times [0,\infty)$ satisfying $t-t^0<\zeta$ the following inequality holds

Figures (28)

  • Figure 1: Age-wise distribution of confirmed dengue cases in Brazil. Number of confirmed dengue cases per 1 million people across various age groups from 2021 to 2024, reported on a weekly basis. The data were obtained from the Brazilian Ministry of Health’s official database Brazil_source.
  • Figure 2: Flow chart of the proposed model.
  • Figure 3: Age-wise population distribution of Pakistan and Japan. Distribution of the population across various age groups in Pakistan and Japan for the year 2025.
  • Figure 4: Illustration of the three possible relations between basic effective reproduction number $R_{\text{eff}}^0$ and growth rate $\lambda^*$.
  • Figure 5: Time varying death rate due to dengue in Brazil. The Subfigure present the death rate due to dengue per week for child susceptible (a), for adult susceptible (b), for middle-aged adult susceptible(c), and for elderly susceptible (d) in Brazil from 2021 to 2024. The solid gold line represents the mean value of the different age groups, while the black dashed line shows the values estimated by the particle filter. The sky blue colored region indicates the 95% confidence interval of these estimated values.
  • ...and 23 more figures

Theorems & Definitions (11)

  • Remark 2.2: Idealistic population
  • Lemma 4.1
  • proof
  • Remark 5.1: Basic reproduction number
  • Lemma 6.1
  • proof
  • Theorem 7.1
  • proof
  • Remark 11.1
  • Lemma 15.1
  • ...and 1 more