Simulation model for the dynamics of dengue with asymptomatic transmission and the effect of temperature

Julián Alejandro Olarte García; Steven Raigosa Osorio; Oscar Andrés Manrique Arias; Carlos Alberto Abello Muñoz; Cesar Augusto Acosta Minoli

doi:10.12688/f1000research.109572.1

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Research Article

Simulation model for the dynamics of dengue with asymptomatic transmission and the effect of temperature

[version 1; peer review: 2 not approved]

Julián Alejandro Olarte García ¹, Steven Raigosa Osorio¹, Oscar Andrés Manrique Arias¹, Carlos Alberto Abello Muñoz¹, Cesar Augusto Acosta Minoli²

Julián Alejandro Olarte García ¹, Steven Raigosa Osorio¹, [...] Oscar Andrés Manrique Arias¹, Carlos Alberto Abello Muñoz¹, Cesar Augusto Acosta Minoli²

PUBLISHED 19 May 2022

Author details Author details

¹ Grupo de Modelación Matemática en Epidemiología (GMME), Universidad del Quindío, Armenia, Quindío, 63004, Colombia
² Grupo de Estudio y Desarrollo de Software (GEDES), Universidad del Quindío, Armenia, Quindío, 63004, Colombia

Julián Alejandro Olarte García
Roles: Conceptualization, Formal Analysis, Writing – Review & Editing

Steven Raigosa Osorio
Roles: Conceptualization, Software, Writing – Original Draft Preparation

Oscar Andrés Manrique Arias
Roles: Conceptualization, Software

Carlos Alberto Abello Muñoz
Roles: Methodology, Supervision

Cesar Augusto Acosta Minoli
Roles: Funding Acquisition, Project Administration

OPEN PEER REVIEW

REVIEWER STATUS

This article is included in the Pathogens gateway.

This article is included in the Emerging Diseases and Outbreaks gateway.

Abstract

Background: One of the fastest spreading vector-borne diseases in tropical and subtropical regions is dengue, which generates cost overruns for public health entities. Several factors can influence the dynamics of dengue virus transmission: environmental and climatic (abundance of vectors), interactions between hosts (infections by asymptomatic individuals), and population immunological factors. Given these conditions, it is necessary to carry out theoretical studies based on meteorological factors and asymptomatic transmission that are associated with both the existence of the vector and its incidence, in order to provide a scientific basis for health entities in decision-making.
Methods: A mathematical model based on nonlinear ordinary differential equations is proposed to interpret the dynamics of dengue transmission in humans coupled to the dynamics of the Aedes aegypti species, considering the population of symptomatic and asymptomatic infected humans and the effect of temperature variability. The basic reproduction number was found and some simulation results based on the Runge-Kutta numerical method were obtained.
Results: The simulations showed that the temperature had a directly proportional relationship with the basic reproduction number. The cases of infected people and carrier mosquitoes increased when the temperature peaks increased drastically; in low temperatures the infection persisted with low morbidity due to the survival of asymptomatic people.
Conclusions: High temperatures tolerable by mosquitoes increase their life expectancy and their numbers in the environment which, together with a reservoir of asymptomatic infected people, leads to a higher incidence of the dengue virus in certain seasons or maintains its circulation in seasons of low temperatures, despite lower vector survival rates.

Keywords

Dengue, temperature, Aedes aegypti, asymptomatic, basic reproduction number, simulation model

Corresponding author: Julián Alejandro Olarte García

Competing interests: No competing interests were disclosed.

Grant information: This research received funding from Vicerrectoría de Investigaciones, Universidad del Quindío.
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Copyright: © 2022 Olarte García JA et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Olarte García JA, Raigosa Osorio S, Manrique Arias OA et al. Simulation model for the dynamics of dengue with asymptomatic transmission and the effect of temperature [version 1; peer review: 2 not approved]. F1000Research 2022, 11:539 (https://doi.org/10.12688/f1000research.109572.1) First published: 19 May 2022, 11:539 (https://doi.org/10.12688/f1000research.109572.1) Latest published: 19 May 2022, 11:539 (https://doi.org/10.12688/f1000research.109572.1)

Introduction

Dengue is a viral disease transmitted by mosquitoes of the Aedes genus, mainly by the Aedes aegypti species, characterized by biphasic fever, myalgia, skin rash, among other symptoms, and can evolve into a clinical condition called severe dengue that can lead to death¹; this disease is a public health problem worldwide, since it is estimated that about 390 million people are infected annually, and no effective vaccine is available to reduce its incidence.² As a consequence, the approaches to reduce its transmission are directed towards control of vector transport by humans through land, air and sea transportation, which has caused its successful colonization around the world.³ For this reason, and due to the introduction of infectious people into new territories, the dengue virus has spread to more than 100 countries.¹

Scientifically, it is known that some people infected with dengue are asymptomatic, defined as those infected who do not present clinical symptoms detectable by public health systems; asymptomatic people were considered dead points in the transmission of the dengue virus, since it was believed that their average level of viremia was not high enough to transmit the virus to the mosquito vector, however, recent studies have shown that asymptomatic people, despite their low average level of viremia, are potentially infectious for mosquitoes. It is estimated that of the nearly 390 million cases of dengue infections per year, around 300 million are asymptomatic, that is, no apparent disease interrupts the daily routine of these individuals.⁴^-⁶

This article is organized as follows: the mathematical model that interprets the dynamics of dengue transmission including the asymptomatic human population coupled to the life cycle of Aedes aegypti and temperature variability is formulated in the subsequent sections Methods and Mathematical model formulation; the basic reproduction number of the infection is determined and interpreted in the Basic reproduction number section; the Results and discussion section presents facts from numerical simulations based on the Runge-Kutta approximation method. Finally, the turn of this research considering other factors that affect the transmission dynamics of the dengue virus is discussed in the Conclusions section.

Methods

A mathematical model based on nonlinear ordinary differential equations was formulated to describe the dynamics of dengue virus transmission through the interaction of human populations and female Aedes aegypti mosquitoes. This is an extension of the susceptible-infected-recovered dynamic for the human population and a susceptible-infected type dynamic for the mosquito population. Some parameters sensitive to temperature depend on it and, in turn, the daily temperature varied over time, in order to delimit the effect of temperature changes on the behavior of populations. The system simulations were carried out in Matlab R2020b software, implementing an algorithm of the fourth order using the Runge-Kutta numerical method⁷^,⁸ with values for the constant parameters extracted from the literature.

Mathematical model formulation

The formulated mathematical model is an extension of the susceptible-infected-recovered dynamic (SIR) for the human population and a susceptible-infected (SI)-type dynamic for the mosquito population; a single dengue serotype is assumed in the transmission dynamics; the infected persons capable of transmitting the dengue virus are divided into two states, symptomatic and asymptomatic, and only the population of female mosquitoes is considered; once a mosquito has become a carrier of a dengue virus serotype, it never ceases to be so.

To interpret the dynamics of dengue transmission, it is assumed that the average number of susceptible people S(t) increases constantly at the rate μN and decreases proportionally to the human mortality rate μ; at time t, susceptible persons can become an average number I(t) of infected or an average number A(t) of asymptomatic by having effective contact with the population of carrier mosquitoes at rates λ₁ and λ₂, which represent the incidence of the population of symptomatic people and the population of asymptomatic people, respectively. The rate of change of susceptible persons with respect to time is given by:

(1)

\frac{dS}{dt} = μN - λ_{1} S - λ_{2} S - μS .

Populations of symptomatic and asymptomatic infected persons increase at incidence rates λ₁ and λ₂, respectively, and decrease proportionally to the natural death rate μ, as well as by the rate of persons recovering from a dengue virus serotype and who acquire immunity proportionally to the recovery rate θ; the population of recovered people R(t) decreases proportionally to the natural death rate μ and increases according to the recovery rate θ of the infected populations. The equations that represent the change of the infected asymptomatic, symptomatic people and the recovered population over time are given by:

(2)

\frac{dA}{dt} = λ_{2} S - θA - μA,

(3)

\frac{dI}{dt} = λ_{1} S - θI - μI,

(4)

\frac{dR}{dt} = θI + θA - μR .

To describe the dynamics of the female mosquito population, an intrinsic growth rate ϕ in the population of non-dengue virus-carrying mosquitoes is considered, and the variable U(t) denotes the average number of non-carrier mosquitoes; denoting the average number of dengue virus-carrying mosquitoes by V(t), the rate of change of this variable depends on the proportion of non-carrier mosquitoes that come into effective contact with the population of infected people and become carrier mosquitoes, proportional to the incidence rates λ_I and λ_A, which represent the population incidence of carrier and non-carrier mosquitoes, respectively. Both virus carrier and non-carrier mosquito populations decrease proportionally to the natural death rate of mosquitoes ε(T) which depends on temperature. Therefore, the equations that describe the change in the population of female mosquitoes over time are given by:

(5)

\frac{dU}{dt} = ϕ - (λ_{I} + λ_{A}) U - ϵU,

(6)

\frac{dV}{dt} = (λ_{I} + λ_{A}) U - ϵV .

From the equations described above, the system of nonlinear ordinary differential equations (7) - (12) emerges to interpret the dynamics of the dengue virus, including the population of the Aedes aegypti vector and the human population.

(7)

\frac{dS}{dt} = μN - λ_{1} S - λ_{2} S - μS \equiv f_{1} (\cdot),

(8)

\frac{dA}{dt} = λ_{2} S - θA - μA \equiv f_{2} (\cdot),

(9)

\frac{dI}{dt} = λ_{1} S - θI - μI \equiv f_{3} (\cdot),

(10)

\frac{dR}{dt} = θI + θA - μR \equiv f_{4} (\cdot),

(11)

\frac{dU}{dt} = ϕ - (λ_{I} + λ_{A}) U - ϵU \equiv f_{5} (\cdot),

(12)

\frac{dV}{dt} = (λ_{I} + λ_{A}) U - ϵV \equiv f_{6} (\cdot) .

The terms λ₁ and λ₂ are governed by a standard incidence, where the rate of newly infected people passing from the susceptible state to the symptomatic or asymptomatic infected state is proportional to the total number of effective contacts between susceptible individuals and the proportion of mosquitoes carrying a dengue virus serotype; these terms are described by:

λ_{1} = fαβ \frac{V}{M}; λ_{2} = (1 - f) αβ \frac{V}{M} .

Similarly, the terms $λ_{I}$ and $λ_{A}$ are also governed by a standard incidence. Here the rate of new cases of carrier mosquitoes is proportional to the total number of effective contacts between non-carrier mosquitoes and infected symptomatic or asymptomatic persons, respectively, these terms are described for:

λ_{I} = ασ \frac{I}{N}; λ_{A} = ασ \frac{A}{N} .

The total population of humans and female mosquitoes, described by N = S + A+ I + R and M = U + V respectively, grow at rates:

\frac{dN}{dt} = \frac{dS}{dt} + \frac{dA}{dt} + \frac{dI}{dt} + \frac{dR}{dt}; \frac{dM}{dt} = \frac{dU}{dt} + \frac{dV}{dt} .

These rates are such that:

(13)

\frac{dN}{dt} = 0; \frac{dM}{dt} = ϕ - ϵM .

From the equations (13) it can be deduced that the total population of humans N is constant over time, in addition to the fact that asymptotically, the total population of female mosquitoes tends to $ϕ / ϵ$ .

For the system of equations (7) - (12), the following initial conditions were considered: $S (0) = S_{0}, A (0) = A_{0}, I (0) = I_{0}, R (0) = R_{0}, U (0) = U_{0}, V (0) = V_{0}$ , as well as the following restrictions for the parameters: $0 < α, 0 < θ, 0 < ϵ, 0 < μ, 0 < ϕ, 0 < f < 1, 0 < β < 1, 0 < σ < 1$ . The region of epidemiological sense is given by:

Ω = \{(S, A, I, R, U, V) \in ℝ_{+}^{6}| 0 < S + A + I + R \leq N| 0 \leq U + V \leq \frac{ϕ}{ϵ}\} .

The temperature-dependent parameters are represented by the following functional expressions described and compiled by authors such as Hyojung Lee et al. in Ref. 9, Liu-Helmersson et al. in Ref. 10 and Polwiang in Ref. 11:

Daily biting rate:

(14)

α (T) = \{\begin{matrix} 0.000202 T (T - 13.35) \sqrt{40.08 - T} & if & {13.35}^{\circ} C < T < {40.08}^{\circ} C \\ 0 & if & otherwise \end{matrix}

Probability of mosquito-to-human transmission:

(15)

β (T) = \{\begin{matrix} 0.001044 T (T - 12.286) \sqrt{32.461 - T} & if & {12.26}^{\circ} C < T < {32.46}^{\circ} C \\ 0 & if & otherwise \end{matrix}

Probability of human-to-mosquito transmission:

(16)

β (T) = \{\begin{matrix} - 0.9037 + 0.0729 T & if & {12.4}^{\circ} C < T < {26.1}^{\circ} C \\ 1 & if & {26.1}^{\circ} C < T < {32.5}^{\circ} C \\ 0 & if & otherwise \end{matrix}

Natural death rate of the mosquito:

(17)

ϵ (T) = 8.692 \times 10^{- 1} - 1.590 \times 10^{- 1} T + 1.116 \times 10^{- 2} T^{2} - 3.408 \times 10^{- 4} T^{3} + 3.809 \times 10^{- 6} T^{4}

The description and the value of the parameters used for the construction of the model are shown in the Table 1.

Table 1. Parameter values.

Parameters	Definition	Value	References
μ	Mortality rate of the human population	0.000039	¹⁴^,¹⁵
θ	Recovery rate of a dengue virus serotype	0.14-0.33	¹⁶^,¹⁷
φ	Constant increase in the population of female mosquitoes	400-5000	¹⁶^-¹⁸
f	Fraction of people who acquire the virus and are symptomatic	0-1	---
α(T)	Daily biting rate	Variable	⁹^-¹¹
β(T)	Probability of transmission from mosquito to human by bite	Variable	⁹^-¹¹
σ(T)	Probability of transmission from human to mosquito by bite	Variable	⁹^-¹¹
ε(T)	Natural death rate of mosquitoes	Variable	⁹^-¹¹

Basic reproduction number

In the epidemiology of infectious diseases, the basic reproduction number (R₀) is one of the most relevant quantitative parameters in situations of epidemic outbreaks. As a biological definition, the basic reproduction number R₀ is said to be the number of secondary cases produced by an infected individual introduced into a fully susceptible population during its entire infectious period. Mathematically, the R₀ is characterized by considering the infectious process as a demographic process in which the increase in offspring is not considered as a birth, but rather as the introduction of new cases of infected through transmission, which entails to look at transmission in terms of consecutive generations of infected individuals.¹²^,¹³

Since R₀ is considered a disease threshold, its value determines whether or not an epidemic exists: if R₀ > 1, an infected person will infect more than one susceptible person during their infectious period and the disease will spread, generating an epidemic; otherwise, if R₀ < 1 the disease will tend to disappear, as this indicates that an infected person introduced into a totally susceptible population transmits the disease on average to less than one person, so the disease will tend to become extinct over time.¹³^,¹⁴

Considering the methodology of the spectral radius of the matrix of the next generation,¹⁹ the equilibrium point without the presence of the disease $E_{0} = (N, 0, 0, 0, \frac{ϕ}{ϵ}, 0)$ and the asymptotic behavior of the total population of mosquitoes M, we proceeded to calculate the basic reproduction number R₀, which is determined by the following expression:

(18)

R_{0} = \frac{(1 - f) α^{2} σβ}{ϵ (θ + μ)} + \frac{f α^{2} σβ}{ϵ (θ + μ)}

In order to have a clearer conception of what the expression of (18) represents, the terms that compose it must be interpreted. First, the additive effect that represents one of the considerations of the model can be appreciated, since it takes into account two routes or sources of infection caused by the population of asymptomatic and symptomatic infected people: the expression $\frac{(1 - f) αβ}{θ + μ}$ in the first additive term represents the incidence of new cases of asymptomatic infections in the human population; on the other hand, the expression $\frac{fαβ}{θ + μ}$ in the second additive term represents the incidence of new cases of asymptomatic infections in the human population, and the expression $\frac{ασ}{ϵ}$ shared by the two additive terms represents the new cases of mosquitoes carrying the virus.

After having an expression for the basic reproduction number based on the parameters considered in the model, R₀ was simulated taking the values included in Table 1; for the temperature-dependent parameters, the daily temperature data collated by the meteorological forecasting company AccuWeather Inc.²⁸ were used and the functions (14) - (17) were evaluated for the mean value between the minimum and the maximum daily temperatures of the city of Armenia - Quindío, Colombia, as shown in Figure 1.

Figure 1. Basic reproduction number.

Figure 1 shows the graphical representation of the basic reproduction number (R₀) throughout a year, compared with the daily temperature data for the year 2019. First, highlighting the biological definition of R₀, it can be seen that most of the time considered in the simulation, the value of the basic reproduction number exceeds the epidemic level 1, from which it can be deduced that throughout the year there was a continuous epidemic risk. Another result of importance observed is the effect produced by changes in temperature over time, since comparing the continuous red line that represents R₀ and the blue dotted line that represents daily temperature, the increase in temperature produces in R₀ a increase in its value, from which it can be deduced that the increase in temperature increases the risk of producing an epidemic of dengue disease in the city of Armenia.

Results and discussion

The simulations of the system (7) - (12) are carried out in Matlab R2020b software, implementing an algorithm of the fourth order Runge Kutta numerical method⁷^,⁸ with values for the parameters retrieved from the literature and described in Table 1. The following simulations show the variation over time of the populations considered and the variation over time of the daily temperature, in order to demarcate the effect of temperature changes on the behavior of the populations. Figure 2 shows the behavior of the symptomatic and asymptomatic infected population, considering that the fraction of people who acquire the virus and are symptomatic is f = 0.75, which means that a quarter of those infected have no symptoms.

Figure 2. Population of symptomatic and asymptomatic infected people with f = 0.75.

From Figure 2 it appears that when considering a lower fraction of asymptomatic infected than of symptomatic infected people, the population of symptomatic people is higher than that of asymptomatic people; however, the population of asymptomatic people permanently marked the evolution of the symptomatic class throughout the simulation, since asymptomatic people are those infected with the dengue virus with the ability to transmit it despite not presenting any type of clinical symptomatology. Other important characteristics that could be observed were the high and low peaks of the blue curve in the lower graph of Figure 2, which represents the daily temperature variability in Armenia (Quindío) in the year 2019, are reflected in the time-series of populations of symptomatic and asymptomatic people. From this, it can be deduced that the increase in temperature increases the cases of symptomatic and asymptomatic infected people, and more specifically, that sudden changes in temperature have a significant impact on the increase or decrease of these two populations.²⁰^–²³

The first graph in Figure 3 shows that the population of non-carrier female mosquitoes grew and quickly stabilized; the second graph of Figure 3 shows that the trend of the population of mosquitoes carrying the dengue virus reaches its maximum on day 50 and day 250. From this it can be deduced that a transmission pattern is present during most of the year and the variability of the temperature represented by the blue dotted curve is a significant determinant for the population of carrier mosquitoes, since the tendency of the mosquitoes to increase carriers coincided with temperature peaks. However, it is not as evident as in the behavior of the population of symptomatic and asymptomatic infected in Figure 2. When comparing both graphs, it can be said that the largest population of carrier female mosquitoes and of symptomatic and asymptomatic infected were found in between 50 to 250 days, which seems reasonable, since a greater number of carrier mosquitoes causes a greater number of new infection cases.

Figure 3. Population of carrier and non-carrier female mosquitoes with f = 0.75.

With the aim of determining the effect of the asymptomatic population on the dynamics of dengue transmission, in the following simulations the same parameters considered in previous simulations exposed in Table 1 were used, changing the value of the fraction of people who acquire the virus and were symptomatic to f = 0.25. In other words, it will be considered that the fraction of people who acquire the virus and are asymptomatic is equivalent to three quarters of the total number of people infected with a dengue virus serotype, which has been suggested in the literature.⁴^-⁶^,²⁴ The graphs for the populations of symptomatic and asymptomatic infected people are shown below.

As expected, in Figure 4 the population size of asymptomatic infected people was greater than that of symptomatic people, which is due to a lower value of the fraction of people who acquire the virus and are symptomatic (f = 0.25) than for previous simulations; in this case, it could be interpreted that when considering the suggestions exposed in the literature mentioned in the previous paragraph, most of the infected population are asymptomatic, which causes a great challenge for creating virus transmission control strategies; this population represents a constant reservoir of the virus that is not identified as an infected population in the clinical history (patients usually appear to health professionals after symptoms have already manifested) and interruption of the early cycle of transmission is difficult.²⁵^–²⁷

Figure 4. Population of symptomatic and asymptomatic infected people with f = 0.25.

Given that f and (1− f) are fractions that complement each other and the population of mosquitoes carrying a dengue virus serotype considers both populations in its dynamics (observe equation (12)), the behavior of the population did not present any change with respect to Figure 3 and its inclusion in the simulations using f = 0.25 becomes unnecessary.

Conclusions

We presented a simulation model based on nonlinear ordinary differential equations describing the transmission dynamics of the dengue virus, considering the population of asymptomatic infected people and some entomological parameters that vary depending on the temperature of the city of Armenia (Quindío), Colombia. The basic reproduction number was determined and the simulations were carried out using a temporal a window of one year; the graphs show that the R₀ exceeds the epidemic level 1, from which it is deduced that throughout the year there was a risk of continuous epidemic; in addition, the temperature had a directly proportional relationship with the basic reproduction number, and therefore climatic change increases the risk of a dengue epidemic.

The simulations of the proposed system of equations reveal that the population of asymptomatic people prevails over time, which can be considered as a constant reservoir of the dengue virus; in addition, the cases of infected people and carrier mosquitoes increased when the peaks of temperature increased drastically, a phenomenon which could be explained by the idea that the city of Armenia has a temperate climate, for which rain leads to a decrease in temperature, which results in clean water deposits; this in turn favors the reproduction of the vector as temperatures rise, increasing the average number of mosquitoes in the environment. Combined with a reservoir of asymptomatic infected people, this leads to a higher incidence of dengue virus.

In the future, the effect of human population movements will be determined considering that, as mentioned in this paper, the asymptomatic population has the capacity to transmit the dengue virus and its condition facilitates the transport of the virus to different areas of the city of Armenia (Quindío) Colombia. Since this population presents no symptoms, preventive measures are not taken to avoid the transmission of the virus.

Data availability

Underlying data

Zenodo: Daily temperature data of the city of Armenia, Colombia year 2019, https://doi.org/10.5281/zenodo.6328228.²⁹

This project contains the following underlying data:

- Daily temperature data of the city of Armenia, Colombia year 2019.csv

Extended data

Zenodo: Parsing Code, https://doi.org/10.5281/zenodo.6328289.³⁰

This project contains the following extended data:

Data are available under the terms of the Creative Commons Attribution 4.0 International license (CC-BY 4.0).

Archived analysis code as at time of publication: https://doi.org/10.5281/zenodo.6328289

License: Creative Commons Attribution 4.0 International

Acknowledgements

The authors would like to thank the reviewers for their careful reading of this manuscript and for their comments and suggestions that improved it. They also thank the Vicerrectoría de Investigaciones of the Universidad del Quindío for taking responsibility for the APC of this publication and the Grupo de Modelación Matemática en Epidemiología of the Universidad del Quindío for being an integral part of the authors’ scientific practice.

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Author details Author details

¹ Grupo de Modelación Matemática en Epidemiología (GMME), Universidad del Quindío, Armenia, Quindío, 63004, Colombia
² Grupo de Estudio y Desarrollo de Software (GEDES), Universidad del Quindío, Armenia, Quindío, 63004, Colombia

Julián Alejandro Olarte García
Roles: Conceptualization, Formal Analysis, Writing – Review & Editing

Steven Raigosa Osorio
Roles: Conceptualization, Software, Writing – Original Draft Preparation

Oscar Andrés Manrique Arias
Roles: Conceptualization, Software

Carlos Alberto Abello Muñoz
Roles: Methodology, Supervision

Cesar Augusto Acosta Minoli
Roles: Funding Acquisition, Project Administration

Competing interests

No competing interests were disclosed.

Grant information

This research received funding from Vicerrectoría de Investigaciones, Universidad del Quindío.
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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Published: 19 May 2022, 11:539

https://doi.org/10.12688/f1000research.109572.1

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Olarte García JA, Raigosa Osorio S, Manrique Arias OA et al. Simulation model for the dynamics of dengue with asymptomatic transmission and the effect of temperature [version 1; peer review: 2 not approved]. F1000Research 2022, 11:539 (https://doi.org/10.12688/f1000research.109572.1)

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Version 1

VERSION 1

PUBLISHED 19 May 2022

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Reviewer Report 16 Nov 2023

J H Arias-Castro, Universidad del Valle, Cali, Colombia

Not Approved

https://doi.org/10.5256/f1000research.121086.r205738

The article aims to analyze the effects of temperature on dengue transmission considering the asymptomatic population. Initially, a model in which some temperature-dependent parameters are considered is presented. But then, the classical analysis of the model is performed without considering ... Continue reading

The article aims to analyze the effects of temperature on dengue transmission considering the asymptomatic population. Initially, a model in which some temperature-dependent parameters are considered is presented. But then, the classical analysis of the model is performed without considering the dependence of the parameters on temperature, which simplifies the analysis of the model and puts it in the classical scheme, which practically makes the subject to be treated lose novelty. Additionally, some scenarios are presented in Figures 3 and 4, which turn out to be analogous because they model situations that have no differences, since the equations turn out to be equivalent, in the case of asymptomatic and infected humans.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Dynamical systems, mathematical epidemiology, biomathematics, optimal control theory

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

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3

Reviewer Report 16 Nov 2023

Sangeeta Saha, Wilfrid Laurier University, Waterloo, Ontario, Canada

Not Approved

https://doi.org/10.5256/f1000research.121086.r212410

The authors, in the manuscript, have proposed a compartmental epidemic model of dengue transmission where the mosquito biting rate and the transmission rates from host to vector as well as vector to host are assumed to be temperature dependent. The ... Continue reading

The authors, in the manuscript, have proposed a compartmental epidemic model of dengue transmission where the mosquito biting rate and the transmission rates from host to vector as well as vector to host are assumed to be temperature dependent. The calculations are basic ones and seem to be ok, but there are few points which I need to mention. Firstly, whenever a model is proposed, it is very important to show the biological well-posedness of the system. So, proving the non-negativity and boundedness of the system variables make the base on which the rest of the analysis is performed. Secondly, I am unable to understand how the transmission from mosquito to human depends on the temperature with two types of conditions (noted in equations 15 and 16). It could have been analysed appropriately. Moreover, it is not demonstrated properly how the time variable is connected with the temperature. So, a proper analysis of the second subfigures of each of Figure 2- Figure 4 could improve the work. Also, as per the model assumption, the parameter denoting 'the increase in female mosquito population' should also depend on temperature, but it is chosen as a constant value only. The reason supporting it needs to be mentioned. Altogether I have found the concept interesting, but the mentioned points, if taken care of, will make the work more strong and presentable only.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

Partly
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Mathematical Biology, Nonlinear Dynamics, Ecological Modelling, Epidemiological Modelling

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

CITE

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VERSION 1 PUBLISHED 19 May 2022

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Version 1 19 May 22	read	read

Sangeeta Saha, Wilfrid Laurier University, Waterloo, Canada
J H Arias-Castro, Universidad del Valle, Cali, Colombia

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Reviewer Report

1 Views

16 Nov 2023 | for Version 1

J H Arias-Castro, Universidad del Valle, Cali, Colombia

1 Views Cite this report Responses(0)

Not Approved

The article aims to analyze the effects of temperature on dengue transmission considering the asymptomatic population. Initially, a model in which some temperature-dependent parameters are considered is presented. But then, the classical analysis of the model is performed without considering the dependence of the parameters on temperature, which simplifies the analysis of the model and puts it in the classical scheme, which practically makes the subject to be treated lose novelty. Additionally, some scenarios are presented in Figures 3 and 4, which turn out to be analogous because they model situations that have no differences, since the equations turn out to be equivalent, in the case of asymptomatic and infected humans.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Dynamical systems, mathematical epidemiology, biomathematics, optimal control theory

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

Respond to this report

Responses (0)

Back to all reports

Reviewer Report

3 Views

16 Nov 2023 | for Version 1

Sangeeta Saha, Wilfrid Laurier University, Waterloo, Ontario, Canada

3 Views Cite this report Responses(0)

Not Approved

The authors, in the manuscript, have proposed a compartmental epidemic model of dengue transmission where the mosquito biting rate and the transmission rates from host to vector as well as vector to host are assumed to be temperature dependent. The calculations are basic ones and seem to be ok, but there are few points which I need to mention. Firstly, whenever a model is proposed, it is very important to show the biological well-posedness of the system. So, proving the non-negativity and boundedness of the system variables make the base on which the rest of the analysis is performed. Secondly, I am unable to understand how the transmission from mosquito to human depends on the temperature with two types of conditions (noted in equations 15 and 16). It could have been analysed appropriately. Moreover, it is not demonstrated properly how the time variable is connected with the temperature. So, a proper analysis of the second subfigures of each of Figure 2- Figure 4 could improve the work. Also, as per the model assumption, the parameter denoting 'the increase in female mosquito population' should also depend on temperature, but it is chosen as a constant value only. The reason supporting it needs to be mentioned. Altogether I have found the concept interesting, but the mentioned points, if taken care of, will make the work more strong and presentable only.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

Partly
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Mathematical Biology, Nonlinear Dynamics, Ecological Modelling, Epidemiological Modelling

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

Respond to this report

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[2] 2. Organización Mundial de la Salud (OMS):Dengue.2017. Reference Source

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[9] 9. Lee H, Kim JE, Lee S, et al.: Potential effects of climate change on dengue transmission dynamics in Korea. PloS one. 2018; 13(6): e0199205. PubMed Abstract | Publisher Full Text

[10] 10. Liu-Helmersson J, Stenlund H, Wilder-Smith A, et al.: Vectorial capacity of Aedes aegypti: effects of temperature and implications for global dengue epidemic potential. PloS one. 2014; 9(3): e89783. PubMed Abstract | Publisher Full Text

[11] 11. Polwiang S: The seasonal reproduction number of dengue fever: impacts of climate on transmission. PeerJ. 2015; 3: e1069. PubMed Abstract | Publisher Full Text

[12] 12. Diekmann OJ, Heesterbeek AP, Metz JAJ: On the definition and the computation of the basic reproduction ratio R₀ in models for infectious diseases in heterogeneous populations. J. Math. Biol. 1990; 28: 365–382. PubMed Abstract

[13] 13. Diekmann O, Heesterbeek JAP, Roberts MG: The construction of next-generation matrices for compartmental epidemic models. J. R. Soc. Interface. 2009; 7(47): 873–885. PubMed Abstract | Publisher Full Text

[14] 14. Chávez JP, Götz T, Siegmund S, et al.: An SIR-Dengue transmission model with seasonal effects and impulsive control. Math. Biosci. 2017; 289: 29–39. PubMed Abstract | Publisher Full Text

[15] 15. Enduri MK, Jolad S: Dynamics of dengue disease with human and vector mobility. Spatial and spatiotemporal epidemiology. 2018; 25: 57–66. PubMed Abstract | Publisher Full Text

[16] 16. Andraud M, Hens N, Marais C, et al.: Dynamic epidemiological models for dengue transmission: a systematic review of structural approaches. PloS one. 2012; 7(11): e49085. PubMed Abstract | Publisher Full Text

[17] 17. Garba SM, Gumel AB, Bakar MA: Backward bifurcations in dengue transmission dynamics. Math. Biosci. 2008; 215(1): 11–25. PubMed Abstract | Publisher Full Text

[18] 18. Sardar T, Rana S, Chattopadhyay J: A mathematical model of dengue transmission with memory. Commun. Nonlinear Sci. Numer. Simul. 2015; 22(1-3): 511–525. Publisher Full Text

[19] 19. Van den Driessche P, Watmough J: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 2002; 180(1-2): 29–48. PubMed Abstract | Publisher Full Text

[20] 20. Watts DM, Burke DS, Harrison BA, et al.: Effect of temperature on the vector efficiency of Aedes aegypti for dengue 2 virus. Am. J. Trop. Med. Hyg. 1987; 36(1): 143–152. PubMed Abstract | Publisher Full Text

[21] 21. Lambrechts L, Paaijmans KP, Fansiri T, et al.: Impact of daily temperature fluctuations on dengue virus transmission by Aedes aegypti. Proc. Natl. Acad. Sci. 2011; 108(18): 7460–7465. PubMed Abstract | Publisher Full Text

[22] 22. Carrington LB, Armijos MV, Lambrechts L, et al.: Fluctuations at a low mean temperature accelerate dengue virus transmission by Aedes aegypti. PLoS Negl. Trop. Dis. 2013; 7(4): e2190. PubMed Abstract | Publisher Full Text

[23] 23. Carrington LB, Seifert SN, Armijos MV, et al.: Reduction of Aedes aegypti vector competence for dengue virus under large temperature fluctuations. Am. J. Trop. Med. Hyg. 2013; 88(4): 689–697. PubMed Abstract | Publisher Full Text

[24] 24. Bhatt S, et al.: The global distribution and burden of dengue. Nature. 2013; 496(7446): 504–507. PubMed Abstract | Publisher Full Text

[25] 25. Beckett CG, Kosasih H, Faisal I, et al.: Early detection of dengue infections using cluster sampling around index cases. Am. J. Trop. Med. Hyg. 2005; 72(6): 777–782. PubMed Abstract | Publisher Full Text

[26] 26. Yoon IK, Rothman AL, Tannitisupawong D, et al.: Underrecognized mildly symptomatic viremic dengue virus infections in rural Thai schools and villages. J. Infect. Dis. 2012; 206(3): 389–398. PubMed Abstract | Publisher Full Text

[27] 27. Gordon A, Kuan G, Mercado JC, et al.: The Nicaraguan pediatric dengue cohort study: incidence of inapparent and symptomatic dengue virus infections, 2004–2010. PLoS Negl. Trop. Dis. 2013; 7(9): e2462. PubMed Abstract | Publisher Full Text

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Simulation model for the dynamics of dengue with asymptomatic transmission and the effect of temperature

Abstract

Keywords

Introduction

Methods

Mathematical model formulation

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

Table 1. Parameter values.

Basic reproduction number

(18)

Figure 1. Basic reproduction number.

Results and discussion

Figure 2. Population of symptomatic and asymptomatic infected people with f = 0.75.

Figure 3. Population of carrier and non-carrier female mosquitoes with f = 0.75.

Figure 4. Population of symptomatic and asymptomatic infected people with f = 0.25.

Conclusions

Data availability

Underlying data

Extended data

Acknowledgements

References

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