To determine the maximum equilibrium prevalence of mosquito-borne microparasitic infections, this paper proposes a general model for vector-borne infections which is flexible enough to comprise the dynamics of a great number of the known diseases transmitted by arthropods. From equilibrium analysis, we determined the number of infected vectors as an explicit function of the model’s parameters and the prevalence of infection in the hosts. From the analysis, it is also possible to derive the basic reproduction number and the equilibrium force of infection as a function of those parameters and variables. From the force of infection, we were able to conclude that, depending on the disease’s structure and the model’s parameters, there is a maximum value of equilibrium prevalence for each of the mosquito-borne microparasitic infections. The analysis is exemplified by the cases of malaria and dengue fever. With the values of the parameters chosen to illustrate those calculations, the maximum equilibrium prevalence found was 31% and 0.02% for malaria and dengue, respectively. The equilibrium analysis demonstrated that there is a maximum prevalence for the mosquito-borne microparasitic infections.
Vector-borne diseases such as malaria, dengue, yellow fever, plague, trypanosomiasis, and leishmaniasis have been major causes of morbidity and mortality through human history [
Currently, half of the world’s population is infected with at least one type of vector-borne pathogens [
In the 17th through early 20th centuries, human morbidity and mortality due to vector-borne diseases outstripped all other causes combined [
The historical paradigm of mosquito-borne infections, malaria, accounts for the most deaths than any other human vector-borne diseases, with approximately 300 million people infected and up to one million deaths every year [
Vectors of human diseases are typically species of mosquitoes that are able to transmit viruses, bacteria, or parasites to humans and other warm-blooded hosts [
Approximately 80 percent of vector-borne disease transmission typically occurs among 20 percent of the host populations [
This paper proposes a general model for vector-borne infections which is flexible enough to comprise the dynamics of some known diseases transmitted by arthropods. From equilibrium analysis, we determined the number of infected vectors as an explicit function of the model’s parameters and the prevalence of infection in the hosts. From the analysis, it is also possible to derive the basic reproduction number and the equilibrium force of infection as a function of those parameters and variables. From the force of infection, we were able to conclude that, depending on the disease’s structure and the model’s parameters, there is a maximum value of equilibrium prevalence for each mosquito-borne microparasitic infections. This is important because of the following: (a) knowing the maximum prevalence at equilibrium (neglecting seasonal variations, see below), we can calculate the maximum force of infection and, therefore, the maximum probability that a visitor gets the disease when visiting an endemic region; (b) the maximum force of infection can be immediately used to calculate the maximum incidence of the disease (again neglecting seasonal variations) in an affected endemic region; (c) if a vector-borne disease is introduced in an unaffected area, we can predict the maximum prevalence at equilibrium for the demographic and disease-related parameters in that area. This gives a very good idea of the amount of resources that will be needed to care for these cases. This allows public health authorities to anticipate the (economic and social) importance of a given disease in order to increase public health preparedness to deal with such challenges.
The model we present in this paper is a deterministic model. However, as we will show, it is possible to introduce some elements of stochasticity.
The whole analysis is exemplified by the cases of malaria and dengue fever.
The model that is used to calculate the efficiency of control strategies can be found in [
The populations involved in the transmission are human hosts, mosquitoes, and their eggs. For the purposes of this paper, the term “eggs” also includes the intermediate stages, such as larvae and pupae. Therefore, the population densities are divided into the compartments described in Table
Model variables and their biological meanings.
Variable | Biological meaning |
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Susceptible humans density |
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Latent humans density |
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Infectious humans density |
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Recovered humans density |
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Uninfected mosquitoes density |
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Latent mosquitoes density |
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Infectious mosquitoes density |
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Uninfected eggs (imm. stages) density |
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Infected aquatic forms density |
We first write down the model equations and then explain the meanings of their terms. The model equations are
and
As mentioned above, seasonal influence in the mosquito population was not considered in this paper. The reason for this is that including seasonal variation, that is, considering
The model’s parameters are described in Table
Model’s parameters and their biological significance.
Parameter | Biological meaning |
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Average daily rate of biting (see text) |
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Fraction of bites actually infective to humans |
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Loss of immunity rate |
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Latency rate in humans |
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Loss of infectiousness in humans |
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Human natural mortality rate |
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Birth rate of humans |
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Carrying capacity of humans |
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Disease mortality in humans |
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Human recovery rate |
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Hatching rate of susceptible eggs |
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Latency rate in mosquitoes |
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Natural mortality rate of mosquitoes |
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Oviposition rate |
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Proportion of infected eggs |
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Carrying capacity of eggs |
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Natural mortality rate of eggs |
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Fraction of bites actually infective to mosquitoes |
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Climatic factor |
Let us explain the meaning and limitations of the above model. First, as mentioned, the variables are densities, that is, number of humans/vectors per unit area. Therefore, to use the model as it is written above, we should consider an area where the populations are approximately homogenously distributed and multiply each variable by this area. One particular important point is raised by the term
Let us explain the meaning of this term. The parameter
Therefore, the number of susceptible humans that get the infection per unit time from infected mosquitoes is
Analogously, the term
The term
The terms
represent the birth rate per unit area of susceptible and infected eggs, respectively. Note that we assume that eggs can be born infected, a phenomenon called vertical transmission in the literature.
The term
The other terms are transition terms between the compartments as explained, for example, in [
Model (
Model’s structure as a function of the parameters.
Model’s structure |
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SI | → |
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SIS | → |
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SIR | → |
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SIRS | → |
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SEIR |
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SEIRS |
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The model can also include vaccination, for instance, against yellow fever or dengue, but this subject will not be treated in this work.
From system (
Replacing the values of
Plot of
The expressions for
which is the equilibrium prevalence of the infection in humans and where
The calculation of the total human population expression at equilibrium,
where
From (
Figure
Note that, as expected, the number of infected vectors is a monotonically increasing function of the biting rate.
The force of infection (the incidence density rate) for humans at the equilibrium,
which can be explicitly written in terms of the equilibrium prevalence of the infection in humans (
From (
Hence, depending on the disease, the model’s structure will determine whether the maximum equilibrium prevalence
is large or small. For instance, in a SEIR model such as dengue, where the recovery rate,
Some theoretical vector-borne infections and their maximum equilibrium prevalences in humans as a function of the recovery rate,
Let us illustrate the theory above by comparing two very distinct vector-borne infections, namely, malaria and dengue. In Table
Parameters’ values that determine the maximum equilibrium prevalences of Malaria and Dengue.
Parameter | Malaria | Dengue |
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0.10 days−1 | 0.00 days−1 |
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0.07 days−1 | 0.14 days−1 |
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0.00 days−1 | 0.00 days−1 |
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10−3 days−1 | 10−5 days−1 |
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0.14 days−1 | 0.20 days−1 |
By applying the parameters’ values above to (
Finally, a comment on an important aspect of (
In addition, when
Since the seminal work by Ronald Ross, mathematical models have provided a great deal of theoretical support for understanding the complex dynamics of vector-borne infections, in addition to the important role those models have played in designing and assessing control strategies [
In this work, we propose a general, although very sketchy, model that considers a great deal of the aspects related to the dynamics of mosquito-borne microparasites. From the equilibrium analysis, we calculated the prevalence of the infection in the host populations, from which the number of infected vectors was deduced. In addition, we deduced an explicit expression for the basic reproduction number and the equilibrium force of infection. It was possible then to demonstrate that, provided an equilibrium is reached, each mosquito-borne microparasite has a maximum host prevalence, depending on the disease’s structure and on the value of the parameters. This analysis was exemplified by the calculation of the maximum equilibrium prevalence of malaria and dengue. Once the disease’s structure is determined and the values of the parameters are known, it is possible to calculate the maximum equilibrium prevalence of a mosquito-borne microparasitic infection.
It may be argued that malaria is not exactly a good example of a microparasitic infection. However, although malaria can behave sometimes as a microparasite and sometimes as a macroparasite [
Another important limitation of our approach is that, in order to calculate the equilibrium densities of each of the model’s variables, we have to neglect seasonal fluctuations, which can be very important in the transmission dynamics of such infections like dengue. However, seasonality in some tropical areas is not too important and the results can be applied to the average trend in prevalence levels.
At first inspection, (
The calculation of a maximum value of equilibrium prevalence for a mosquito-borne microparasitic infection may help public health authorities to estimate the resources needed to care for the infected individuals, anticipating the importance of the disease and increasing the public health preparedness to deal with such a challenge.
In this appendix, we present a summary of the sensitivity analysis (explored in detail in [
Results of the sensitivity analysis. The results represent the relative amount of variation (expressed in percentual variation) in the variable if we vary the parameters by 1% (see [
Parameter | Mean |
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Sensitivity of |
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1.94 |
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0.69 |
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(−) |
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(−) 2.42 |
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Sensitivity of |
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5.02 |
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2.32 |
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(−) |
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(−) 5.40 |
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Sensitivity of |
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2.67 |
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1.34 |
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(−) |
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(−) 3.20 |
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research from which these results were obtained has received funding from the European Union’s Seventh Framework Programme (FP7/2007–2013) under Grant agreement no. 282589, from LIM01 HCFMUSP and CNPq. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the paper.