The Ross–Macdonald Model

🦟🦟 The Silent Architects of Epidemics: Unlocking the Ross–Macdonald Model

An Accessible Guide to Vector-Borne Disease Dynamics for Modelers, Public Health Experts, and Global Health Advocates

🔍 Introduction

While viruses like influenza or SARS-CoV-2 spread directly from person to person, a vast array of deadly pathogens—from malaria and dengue to Zika and chikungunya—rely on an invisible intermediary: the mosquito. Understanding these diseases demands more than tracking human cases; it requires modeling a two-host dance between humans and vectors.

Enter the Ross–Macdonald model, the foundational framework for vector-borne disease transmission. First conceptualized by Sir Ronald Ross in 1911 to explain malaria dynamics in India [1], and later refined by George Macdonald in the 1950s [2], this elegant system of ordinary differential equations (ODEs) captures the essential feedback loop between infected humans and infectious mosquitoes.

Despite its century-old origins, the Ross–Macdonald model remains astonishingly relevant—powering everything from WHO malaria elimination strategies to real-time dengue forecasting in Southeast Asia [3–5]. In this article, we unpack its core structure, parameters, and assumptions, then explore how modern variants have expanded its reach to address climate change, insecticide resistance, and spatial heterogeneity.

Whether you’re designing bed net campaigns or simulating urban arboviral outbreaks, mastering the Ross–Macdonald model is your first step toward taming vector-borne threats.

🧩 Model Description

The classic Ross–Macdonald model divides the population into two interacting species:

  • Humans (host):
    • Sₕ(t): Susceptible humans
    • Iₕ(t): Infected (and infectious) humans
    • Total human population: Nₕ = Sₕ + Iₕ (assumed constant)
  • Mosquitoes (vector):
    • Sᵥ(t): Susceptible mosquitoes
    • Iᵥ(t): Infectious mosquitoes
    • Total mosquito population: Nᵥ = Sᵥ + Iᵥ (often assumed constant or seasonally varying)

Transmission occurs in two steps:

  1. A susceptible mosquito bites an infected human and becomes infected.
  2. An infectious mosquito bites a susceptible human and transmits the pathogen.

The dynamics are governed by the following ODE system:

dSₕ/dt = −a · b · (Iᵥ / Nᵥ) · Sₕ

dIₕ/dt = a · b · (Iᵥ / Nᵥ) · Sₕ − r · Iₕ

dSᵥ/dt = −a · c · (Iₕ / Nₕ) · Sᵥ

dIᵥ/dt = a · c · (Iₕ / Nₕ) · Sᵥ − μ · Iᵥ

💡 Key Insight:

  • The term a · (Iᵥ / Nᵥ) is the force of infection on humans: bite rate × proportion of infectious mosquitoes.
  • The term a · (Iₕ / Nₕ) is the force of infection on mosquitoes: bite rate × proportion of infected humans.

Because Nₕ and Nᵥ are constant, we often work with prevalence variables:

  • Human infection prevalence: iₕ = Iₕ / Nₕ
  • Mosquito infection prevalence: iᵥ = Iᵥ / Nᵥ

This leads to the reduced system:

diₕ/dt = a · b · iᵥ · (1 − iₕ) − r · iₕ

diᵥ/dt = a · c · iₕ · (1 − iᵥ) − μ · iᵥ

📊 Parameter Definitions

aBiting rateAverage number of human bites per mosquito per day0.1 – 1.0day⁻¹
bTransmission efficiency (v→h)Probability a bite from infectious mosquito infects a human0.1 – 0.8dimensionless
cTransmission efficiency (h→v)Probability a mosquito becomes infected after biting an infected human0.1 – 0.5dimensionless
rHuman recovery rate1 / average infectious period in humans1/200 – 1/50 (malaria: chronic)day⁻¹
μMosquito mortality rate1 / average mosquito lifespan1/30 – 1/7day⁻¹
NₕHuman populationTotal number of humans in the area10³ – 10⁶persons
NᵥMosquito populationTotal number of female mosquitoes10⁴ – 10⁷ (highly variable)mosquitoes

🌡️ Note: The basic reproduction number for this system is:
R₀ = (a² · b · c · Nᵥ) / (r · μ · Nₕ)
This reveals that reducing mosquito density (Nᵥ) or lifespan (1/μ) has a quadratic impact on transmission—a key insight behind insecticide-treated nets (ITNs) and indoor residual spraying (IRS).

Initial conditions typically assume a small introduction:
Iₕ(0) = 1, Iᵥ(0) = 0 (or vice versa), with Sₕ(0) ≈ Nₕ, Sᵥ(0) ≈ Nᵥ.

⚖️ Assumptions and Applicability

The Ross–Macdonald model relies on several simplifying—but powerful—assumptions:

✅ Constant populations: No births, deaths (except mosquito mortality), or migration.
✅ Homogeneous mixing: All humans are equally likely to be bitten; all mosquitoes bite humans at the same rate.
✅ No latency periods: Mosquitoes become infectious immediately after infection (no extrinsic incubation period).
✅ Fixed parameters: No seasonal or behavioral variation in biting or survival.
✅ Single vector species: Ignores co-circulating mosquito types (e.g., Anopheles gambiae vs. An. funestus).

🎯 When to Use the Ross–Macdonald Model

This model is ideal for:

  • Conceptual understanding of vector-borne transmission thresholds
  • Back-of-the-envelope calculations of R₀ and intervention impact
  • Teaching core principles in epidemiology and public health
  • Baseline scenarios for malaria, dengue, or Zika in stable, endemic settings

It is less appropriate for:

  • Diseases with long extrinsic incubation (e.g., West Nile virus)
  • Urban outbreaks with strong spatial clustering
  • Settings with multiple vector species or animal reservoirs (e.g., yellow fever)
  • Scenarios requiring age structure or immunity dynamics

🚀 Model Extensions and Variants

To bridge the gap between theory and reality, researchers have developed numerous Ross–Macdonald extensions:

1. Ross–Macdonald with Extrinsic Incubation Period (EIP)

Purpose: Account for the time between mosquito infection and infectiousness (critical for temperature-sensitive pathogens).

Modification:
Add an exposed mosquito compartment Eᵥ:
dEᵥ/dt = a · c · (Iₕ / Nₕ) · Sᵥ − (μ + δ) · Eᵥ
dIᵥ/dt = δ · Eᵥ − μ · Iᵥ
where δ = rate of becoming infectious (δ = 1 / EIP)

Application: Modeling how climate warming shortens EIP and expands dengue risk zones [6].

2. Seasonal Ross–Macdonald

Purpose: Capture rainfall- and temperature-driven mosquito population cycles.

Modification:
Let Nᵥ(t) = Nᵥ₀ · (1 + ε · sin(2πt/365 + φ)) or model Nᵥ dynamically:
dNᵥ/dt = λ(t) − μ · Nᵥ
where λ(t) is a seasonal recruitment rate.

Application: Predicting malaria peaks in sub-Saharan Africa during rainy seasons [7].

3. Two-Patch (Meta-Population) Ross–Macdonald

Purpose: Model human movement between high- and low-transmission zones (e.g., forest workers returning home).

Modification:
Define human compartments for patch 1 and 2, with commuting rate m:
dIₕ₁/dt = a₁·b·iᵥ₁·(1−iₕ₁) − r·Iₕ₁ − m·Iₕ₁ + m·Iₕ₂
(similar for patch 2)

Application: Understanding malaria persistence in elimination settings due to imported cases [8].

4. Ross–Macdonald with Insecticide Resistance

Purpose: Evaluate how resistance reduces intervention efficacy.

Modification:
Introduce a resistance allele frequency ρ(t) that reduces mosquito mortality from IRS/ITNs:
Effective mortality = μ + (1−ρ)·τ, where τ = additional mortality from insecticides

Application: Optimizing rotation of insecticide classes in West Africa [9].

5. Stochastic Ross–Macdonald

Purpose: Capture extinction risk in low-transmission settings.

Modification:
Implement the model as a continuous-time Markov chain with probabilistic transitions (e.g., using Gillespie algorithm).

Application: Simulating the probability of malaria elimination after mass drug administration [10].

🌟 Conclusion

The Ross–Macdonald model is more than a historical curiosity—it is a living framework that continues to shape global health policy a century after its inception. Its genius lies in distilling a complex ecological system into a few equations that reveal profound truths: that mosquito lifespan matters more than abundance, that small reductions in biting can yield large gains, and that elimination is possible when R₀ is pushed below 1.

Modern variants have enriched this foundation with realism—seasonality, spatial structure, climate sensitivity, and evolutionary dynamics—yet the core insight remains unchanged: to stop a vector-borne disease, you must disrupt the loop between human and vector.

As emerging pathogens like Mayaro virus and urban malaria threaten new populations, the Ross–Macdonald model—and its descendants—will remain indispensable tools in our arsenal. Not because they are perfect, but because they are clear, actionable, and rooted in biological truth.

🌍 Final Thought: In the fight against vector-borne disease, every equation brings us closer to a world where no child dies from a mosquito bite.

📚 References

  1. Ross, R. (1911). The Prevention of Malaria. John Murray.
  2. Macdonald, G. (1957). The Epidemiology and Control of Malaria. Oxford University Press.
  3. Smith, D. L., et al. (2012). Ross, Macdonald, and a theory for the dynamics and control of mosquito-transmitted pathogens. PLoS Pathogens, 8(4), e1002588.
    https://doi.org/10.1371/journal.ppat.1002588
  1. Reiner, R. C., et al. (2013). A systematic review of mathematical models of mosquito-borne pathogen transmission: 1970–2010. Journal of the Royal Society Interface, 10(81), 20120921.
    https://doi.org/10.1098/rsif.2012.0921
  1. World Health Organization. (2023). World Malaria Report 2023.
    https://www.who.int/publications/i/item/9789240086173
  2. Mordecai, E. A., et al. (2017). Thermal biology of mosquito-borne disease. Ecology Letters, 20(10), 1271–1284.
    https://doi.org/10.1111/ele.12825
  3. Paaijmans, K. P., et al. (2010). Influence of climate on malaria transmission depends on daily temperature variation. Proceedings of the National Academy of Sciences, 107(34), 15135–15139.
    https://doi.org/10.1073/pnas.1006422107
  4. Tatem, A. J., et al. (2017). The use of mobile phone data for the estimation of the travel patterns and imported Plasmodium falciparum rates among Zanzibar residents. Malaria Journal, 16, 477.
  5. Churcher, T. S., et al. (2016). Modelling the impact of mosquito behavior on the efficacy of insecticide treated bed nets. bioRxiv.
    https://doi.org/10.1101/085092
  1. Chitnis, N., et al. (2008). A mathematical model for the dynamics of malaria in mosquitoes feeding on a heterogeneous host population. Journal of Biological Dynamics, 2(3), 259–285.
    https://doi.org/10.1080/17513750701769857