Vector-borne diseases
A vector-borne disease is transmitted between hosts through another living organism called a vector. Mosquitoes, ticks, sand flies and fleas are important examples of vectors.
Malaria and dengue are familiar mosquito-borne examples. Lyme disease is transmitted by ticks. The essential modelling idea is different from an ordinary directly transmitted infection: an infectious human usually does not infect another human directly. Transmission passes through the vector population.
Start with the biological transmission cycle
Imagine a susceptible mosquito biting an infectious human. The mosquito may acquire the pathogen. After becoming capable of transmitting it, that mosquito can later bite a susceptible human and infect that person.
This creates a transmission cycle. Host infection helps generate vector infection, and vector infection helps generate host infection.
Why an ordinary SIR model is not enough
For a directly transmitted disease, a simple SIR model can use an infection term such as
\[\beta\frac{S_H I_H}{N_H}.\]This assumes infectious hosts generate infection pressure directly on susceptible hosts.
For a purely vector-borne infection, that is not the main route. A susceptible host is infected by an infectious vector, while a susceptible vector becomes infected by biting an infectious host.
Step 1: create separate host and vector populations
Use the subscript \(H\) for hosts and \(V\) for vectors. A simple host population can be divided into
\[S_H(t),\qquad I_H(t),\qquad R_H(t),\]while a simple vector population can be divided into
\[S_V(t),\qquad I_V(t).\]| Symbol | Meaning |
|---|---|
| \(S_H\) | susceptible hosts |
| \(I_H\) | infectious hosts |
| \(R_H\) | recovered hosts |
| \(S_V\) | susceptible vectors |
| \(I_V\) | infectious vectors |
| \(N_H\) | total host population |
| \(N_V\) | total vector population |
The two populations are coupled
The horizontal arrows show progression within each population. The curved arrows show the important coupling: infectious vectors create host infections, and infectious hosts create vector infections.
Step 2: understand the biting rate
Let \(a\) be the average number of host bites made by one vector per unit time. For example,
\[a=0.5\text{ bites per vector per day}\]means that, averaged across the vector population, each vector makes one host bite every two days.
The biting rate is important because transmission requires contact between a vector and a host.
Step 3: not every bite transmits infection
A bite is an opportunity for transmission, not a guaranteed infection. We therefore distinguish two transmission probabilities:
| Parameter | Meaning |
|---|---|
| \(b\) | probability that a bite by an infectious vector infects a susceptible host |
| \(c\) | probability that a susceptible vector becomes infected when it bites an infectious host |
The two probabilities need not be equal because transmission from vector to host and host to vector are different biological processes.
Step 4: calculate infection pressure on hosts
Suppose the vector population contains \(I_V\) infectious vectors out of \(N_V\) total vectors. Then
\[\frac{I_V}{N_V}\]is the infectious proportion of the vector population.
To understand the host infection rate, combine three ideas:
Let
\[m=\frac{N_V}{N_H}\]be the number of vectors per host. A common simple force of infection on hosts is then
\[\boxed{\lambda_H=a b m\frac{I_V}{N_V}}.\]Since \(m=N_V/N_H\), this can also be written as
\[\lambda_H=ab\frac{I_V}{N_H}.\]The new-host-infection flow is therefore
\[\boxed{\lambda_HS_H}.\]Work through the host infection term
Suppose:
| biting rate | \(a=0.5\) bites per vector per day |
| vectors per host | \(m=2\) |
| infectious vector proportion | \(I_V/N_V=0.10\) |
| vector-to-host transmission probability | \(b=0.20\) |
Then
\[\lambda_H=0.5\times0.20\times2\times0.10=0.02\text{ day}^{-1}.\]Step 5: calculate infection pressure on vectors
Now reverse the direction. A susceptible vector becomes infected when it bites an infectious host.
The infectious fraction of the host population is
\[\frac{I_H}{N_H}.\]If a vector bites at rate \(a\), and the probability of host-to-vector transmission per relevant bite is \(c\), a common simple force of infection on vectors is
\[\boxed{\lambda_V=ac\frac{I_H}{N_H}}.\]The new-vector-infection flow is therefore
\[\boxed{\lambda_VS_V}.\]Step 6: write a simple host-vector model
A simple SIR-type host model coupled to susceptible-infectious vectors can be written as
\[\frac{dS_H}{dt}=-\lambda_HS_H,\] \[\frac{dI_H}{dt}=\lambda_HS_H-\gamma I_H,\] \[\frac{dR_H}{dt}=\gamma I_H,\]and for vectors,
\[\frac{dS_V}{dt}=\mu_VN_V-\lambda_VS_V-\mu_VS_V,\] \[\frac{dI_V}{dt}=\lambda_VS_V-\mu_VI_V.\]| Parameter | Meaning |
|---|---|
| \(\gamma\) | host recovery rate |
| \(\mu_V\) | vector death rate |
| \(\mu_VN_V\) | simple vector recruitment term maintaining a constant vector population in this example |
These equations are only one simple model. Their purpose here is to show how the two infection processes fit together.
Why vectors are often not given a recovered compartment
In many mosquito-borne disease models, once a mosquito becomes infectious it remains infectious for the rest of its life. It therefore leaves the infectious class through death rather than recovery.
This is why a simple vector model may use only susceptible and infectious vector compartments even when the human population uses SIR compartments.
The vector incubation period
A vector that acquires a pathogen may not become infectious immediately. The pathogen may need time to develop inside the vector. This delay is called the extrinsic incubation period.
To represent it, an exposed-vector compartment can be added:
Now infection of a vector moves it from \(S_V\) to \(E_V\). Only after the incubation period does it enter \(I_V\) and become capable of infecting hosts.
Why vector lifespan matters
The extrinsic incubation period creates an important biological constraint. A vector must survive long enough after becoming infected to reach the infectious stage.
If most vectors die before the pathogen completes development, transmission can be greatly reduced. This is one reason vector mortality can have a strong effect on vector-borne epidemics.
Why vector abundance matters
The ratio
\[m=\frac{N_V}{N_H}\]measures vector abundance relative to the host population. If there are more vectors per host, hosts can receive more vector bites even if the biting behaviour of each individual vector is unchanged.
Why temperature and environment can matter
For many vector-borne diseases, environmental conditions affect the vector and pathogen. Temperature, rainfall, humidity and habitat can influence vector abundance, survival, biting behaviour or pathogen development.
This means parameters such as \(a\), \(N_V\), \(\mu_V\), or the vector incubation rate may vary through time rather than remaining constant.
Host and vector infection curves can be different
Because infection must move between two populations, host and vector prevalence need not rise and peak at exactly the same time.
The figure illustrates a possible lag between host and vector infection. The exact relationship depends on the disease, incubation periods, recovery, vector survival and other parameters.
Interventions act at different points in the cycle
The transmission cycle makes it clear why vector-borne disease control can target several different mechanisms.
| Intervention | Model mechanism affected |
|---|---|
| bed nets or repellents | reduce successful vector-host contact or biting |
| vector control | reduce vector abundance or increase vector mortality |
| removing breeding sites | reduce vector recruitment and future abundance |
| host treatment | shorten infectious duration or reduce host infectiousness |
| vaccination, where available | reduce host susceptibility, disease or transmission depending on vaccine action |
A major advantage of the mathematical model is that these interventions can be connected to the specific biological parameter or transition they change.
The reproduction number is more complicated
For directly transmitted infections, a simple model may produce a formula such as \(R_0=\beta/\gamma\). In a vector-borne system, successful spread depends on the complete host-vector-host cycle.
Consequently, \(R_0\) generally depends on several quantities, including biting rate, vector-to-host and host-to-vector transmission, vector abundance, host infectious duration, vector survival and, when relevant, the probability of surviving the vector incubation period.
Ross and Macdonald
The mathematical study of mosquito-borne transmission is strongly associated with Ronald Ross, whose early twentieth-century work showed mathematically that reducing mosquito density could interrupt malaria transmission. George Macdonald later developed this framework substantially, leading to what is commonly called the Ross–Macdonald model of mosquito-borne disease transmission.
Modern vector-borne models are often much more detailed, but the central host-vector coupling remains fundamental.
Main assumptions of the simple model
| Simple assumption | What reality may contain |
|---|---|
| all hosts are equally exposed | some people receive many more bites than others |
| all vectors bite at the same rate | biting behaviour varies |
| constant vector population | vector abundance can be strongly seasonal |
| constant parameters | temperature and environment can change them |
| random host-vector mixing | vectors and hosts are spatially clustered |
| simple vector infection states | incubation and pathogen development may be important |