🧬 Overview and Conceptual Motivation
For pathogens transmitted through an intermediate organism, transmission dynamics cannot be captured by single-population models. The Host–Vector SEIR–SEI model with latency is a rigorous framework designed to describe diseases such as Dengue, Zika, Malaria, and West Nile virus. The defining feature of this model is the explicit inclusion of Exposed (E) classes in both hosts and vectors. These compartments represent biologically essential delays: the intrinsic incubation period in humans and the extrinsic incubation period in vectors, during which the pathogen matures before onward transmission becomes possible. Accounting for these latent stages is critical for realistic predictions of outbreak timing and intensity.
🏗️ Compartmental Structure and Flow
The model couples two interacting populations: Hosts (H) and Vectors (V), each with its own disease progression.
Host Population (SEIR)
Susceptible Hosts (Sₕ)
Humans who are not infected and are vulnerable to infection through vector bites.
Exposed Hosts (Eₕ)
Humans who have been infected but are not yet infectious.
Infectious Hosts (Iₕ)
Humans capable of transmitting the pathogen to vectors during blood feeding.
Recovered Hosts (Rₕ)
Humans who have cleared the infection and acquired immunity.
Vector Population (SEI)
Susceptible Vectors (Sᵥ)
Vectors that are pathogen-free.
Exposed Vectors (Eᵥ)
Vectors that have ingested infected blood but are not yet infectious.
Infectious Vectors (Iᵥ)
Vectors capable of transmitting the pathogen to susceptible hosts.
Flow structure:
Host: Sₕ → Eₕ → Iₕ → Rₕ
Vector: Sᵥ → Eᵥ → Iᵥ
Transmission occurs bidirectionally between the two populations: infectious vectors infect hosts, and infectious hosts infect vectors.
🧮 Mathematical Formulation
The coupled dynamics of hosts and vectors are described by the following system of ordinary differential equations, including vital dynamics.
Host Dynamics
dSₕ/dt = μₕ · Nₕ − (βₕᵥ · Sₕ · Iᵥ / Nₕ) − μₕ · Sₕ
dEₕ/dt = (βₕᵥ · Sₕ · Iᵥ / Nₕ) − (σₕ + μₕ) · Eₕ
dIₕ/dt = σₕ · Eₕ − (γ + μₕ) · Iₕ
dRₕ/dt = γ · Iₕ − μₕ · Rₕ
Vector Dynamics
dSᵥ/dt = μᵥ · Nᵥ − (βᵥₕ · Sᵥ · Iₕ / Nₕ) − μᵥ · Sᵥ
dEᵥ/dt = (βᵥₕ · Sᵥ · Iₕ / Nₕ) − (σᵥ + μᵥ) · Eᵥ
dIᵥ/dt = σᵥ · Eᵥ − μᵥ · Iᵥ
Here, βₕᵥ and βᵥₕ denote transmission rates between vectors and hosts, σₕ and σᵥ are progression rates through latent stages, μₕ and μᵥ are natural mortality rates, γ is the host recovery rate, and Nₕ and Nᵥ are total host and vector populations.
📋 Table 1. Definition of Model Parameters
| Parameter | Symbol | Definition |
|---|---|---|
| Susceptible hosts | Sₕ | Uninfected human population |
| Exposed hosts | Eₕ | Latently infected humans |
| Infectious hosts | Iₕ | Humans capable of infecting vectors |
| Recovered hosts | Rₕ | Immune humans |
| Susceptible vectors | Sᵥ | Uninfected vectors |
| Exposed vectors | Eᵥ | Latently infected vectors |
| Infectious vectors | Iᵥ | Vectors capable of infecting hosts |
| Host-to-vector transmission | βᵥₕ | Infection rate from host to vector |
| Vector-to-host transmission | βₕᵥ | Infection rate from vector to host |
| Host latency rate | σₕ | Progression from Eₕ to Iₕ |
| Vector latency rate | σᵥ | Progression from Eᵥ to Iᵥ |
| Host recovery rate | γ | Clearance of infection in hosts |
| Mortality rates | μₕ, μᵥ | Natural death rates |
🌤️ Temperature-Dependent Forcing and Environmental Effects
Vector-borne disease dynamics are strongly modulated by temperature because vectors are ectothermic organisms. Many biological processes, including biting frequency, survival, and pathogen development, depend nonlinearly on temperature. The extrinsic incubation rate σᵥ is often modeled as a temperature-dependent function.
A commonly used representation is:
σᵥ(T) = κ · T · (T − Tₘᵢₙ) · (Tₘₐₓ − T)¹ᐟ² for Tₘᵢₙ < T < Tₘₐₓ
In this formulation, Tₘᵢₙ and Tₘₐₓ define thermal limits for pathogen development, and κ is a species-specific scaling constant. As temperature increases within this viable range, the extrinsic incubation period shortens, allowing vectors to become infectious more quickly and increasing overall transmission potential.
📊 Table 2. Realistic Parameter Ranges for Vector-Borne Viral Diseases
| Quantity | Typical Range | Interpretation |
|---|---|---|
| Host latent period (1/σₕ) | 4 – 10 days | Intrinsic incubation in humans |
| Vector latent period (1/σᵥ) | 8 – 12 days | Extrinsic incubation, temperature-dependent |
| Vector lifespan (1/μᵥ) | 10 – 30 days | Survival of adult vectors |
| Biting rate (a) | 0.2 – 0.5 day⁻¹ | Included in transmission coefficients |
| Basic reproduction number (R₀) | Variable | Depends on host–vector coupling |
🎯 Applicability and Limitations
Applicability
This framework is essential for modeling tropical and subtropical diseases transmitted by mosquitoes or other arthropods. It is widely used to evaluate vector control strategies, such as insecticide-treated nets or larviciding, and to assess the impact of vaccination or treatment on host dynamics. The model is also a cornerstone for seasonal forecasting driven by temperature and rainfall patterns.
Key Assumptions and Weaknesses
The simplest formulations assume a constant vector population, whereas real vector densities fluctuate strongly with environmental conditions. Vertical transmission in vectors is typically ignored, despite its relevance for some pathogens. Homogeneous mixing between hosts and vectors is assumed, neglecting spatial clustering and behavioral heterogeneity that create transmission hotspots.
📚 References
- Macdonald, G. (1957). The Epidemiology and Control of Malaria. Oxford University Press.
- Chitnis, N., Cushing, J. M., & Hyman, J. M. (2008). Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bulletin of Mathematical Biology.
- Diekmann, O., et al. (2012). Mathematical Tools for Understanding Infectious Disease Dynamics. Princeton University Press.
- Anderson, R. M., & May, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control. Oxford University Press.
- Mordecai, E. A., et al. (2017). Detecting the impact of temperature on transmission of Zika, dengue, and chikungunya using mechanistic models. PLoS Neglected Tropical Diseases.
🧠 Analogy for Intuition
The Host–Vector SEIR–SEI model can be imagined as a two-factory logistics chain. Hosts are factories producing the pathogen, while vectors are delivery trucks. Latent periods correspond to loading times at each factory. As temperature rises, trucks move faster and loading times shorten, resulting in a sudden surge of infectious deliveries across the system.