## Figures

## Abstract

As temperatures change worldwide, the pattern and competency of disease vectors will change, altering the global distribution of both the burden of infectious disease and the risk of the emergence of those diseases into new regions. To evaluate the risk of potential summer dengue outbreaks triggered by infected travelers under various climate scenarios, we develop an SEIR-type model, run numerical simulations, and conduct sensitivity analyses under a range of temperature profiles. Our model extends existing theoretical frameworks for studying dengue dynamics by introducing temperature dependence of two key parameters: the mosquito extrinsic incubation period and the lifespan of mosquitoes, which empirical data suggests are both highly temperature dependent. We find that changing temperature significantly alters dengue risk in an inverted U-shape, with temperatures in the range 27-31°C producing the highest risk. As temperatures increase beyond 31°C, the determinants of dengue risk begin to shift from mosquito biting rate and carrying capacity to the duration of the human infectious period, suggesting that changing temperatures not only alter dengue risk but also the potential efficacy of control measures. To illustrate the role of spatial and temporal temperature heterogeneity, we select five US cities where the primary dengue vector, the mosquito *Aedes aegypti*, has been observed, and which have had dengue cases in the past: Los Angeles, Houston, Miami, Brownsville, and Phoenix. Our analysis suggests that an increase of 3°C leads to an approximate doubling of the risk of dengue in Los Angeles and Houston, but a reduction of risk in Miami, Brownsville, and Phoenix due to extreme heat.

**Citation: **Trejo I, Barnard M, Spencer JA, Keithley J, Martinez KM, Crooker I, et al. (2023) Changing temperature profiles and the risk of dengue outbreaks. PLOS Clim 2(2):
e0000115.
https://doi.org/10.1371/journal.pclm.0000115

**Editor: **Samuel Nii Ardey Codjoe,
University of Ghana, GHANA

**Received: **January 24, 2022; **Accepted: **December 21, 2022; **Published: ** February 15, 2023

This is an open access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication.

**Data Availability: **All data used in this analysis are publicly available. Details are provided in S1 Table.

**Funding: **IT and NH were funded by the Laboratory Directed Research and Development Program of Los Alamos National Laboratory under project numbers 20210709ER and 20210043DR. IT, MB, JK, KMM, IC, EOR, and CM were funded by the Laboratory Directed Research and Development Program of Los Alamos National Laboratory under project number 20210062DR. NH benefited from the support and resources of the Center for Non-Linear Studies at LANL. JAS was supported by an appointment to the Intelligence Community Postdoctoral Research Fellowship Program at Los Alamos National Laboratory administered by Oak Ridge Institute for Science and Education (ORISE) through an interagency agreement between the U.S. Department of Energy and the Office of the Director of National Intelligence (ODNI). The findings and conclusions in this report are those of the authors and do not necessarily represent the official position of Los Alamos National Laboratory. Los Alamos National Laboratory, an affirmative action/equal opportunity employer, is managed by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under contract 89233218CNA000001. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

**Competing interests: ** The authors have declared that no competing interests exist.

## Introduction

The global reported incidence of Dengue fever, a mosquito-borne disease caused by dengue virus [1], has increased 8-fold in the past two decades. Almost half of the world’s population is now at risk [2], with an estimated 390 million new cases every year [3]. No effective vaccine is broadly available [4], and the primary way to mitigate the disease is through control of or protection from mosquito populations [5–7]. Mosquitoes are a necessary intermediate host for the dengue life-cycle. Mosquitoes are ectothermic, and their life cycle is very sensitive to environmental temperature, which is a key driver of annual dengue case counts [8–13].

The extrinsic incubation period (EIP) is the period from when a mosquito ingests an infectious blood meal to when it becomes infectious (competent to transmit dengue virus). As temperature increases, EIP decreases up to a point, suggesting a proportional relationship between temperature and dengue risk. However, for transmission to take place, the mosquito lifespan, which declines with increasing temperature after a certain point, must exceed the EIP [9, 13–17]. Recent work has quantified the impact of EIP dependence on temperature [14, 18–20].

Considerable modeling work has been done on the effects of temperature dependence of mosquito life traits on arbovirus transmission [21–25]. Mayton et al. established the importance of mosquito age structure on transmission potential, which we include in our model as a temperature dependent lifespan [26]. As temperature changes, the implicit age structure of the mosquito population also changes. Previous models have adopted varied approaches to studying the effects of temperature dependent parameters on disease dynamics. Mordecai et al. integrated multiple temperature dependent parameters to study the relationship of temperature on the epidemic threshold [21], while Kamiya et al. used an explicit epidemic model to examine the effect of temperature dependent EIP while fixing other model parameters [18]. Our study extends these approaches by allowing both lifespan and EIP to vary with temperature in a explicit mechanistic model that both allows us to simulate dynamics and define the epidemic threshold. This approach allows us to characterize temperature-dependent dengue risk more fully by including key drivers of the life-history trade-offs for both dengue and its mosquito hosts. The effects of a changing climate on mosquito-borne disease are not monotonic, and increasing temperatures can have complex effects not only the global distribution of dengue but also on mitigation strategies [13, 18, 21, 27].

Endemic dengue has been historically limited to tropical climates, since *A. aegypti* cannot survive below 8°C [28]. However, single dengue outbreaks initiated by infected travelers have recently been confirmed in temperate regions [10, 29, 30]. Outbreaks are possible in cities with warm summers and mild winters near the northern borders of *A. aegypti* ranges. We have chosen to study the potential for outbreaks in two groups of U.S. cities that together represent a range of mean temperatures from 20.6°C to 31.8°C: (1) those where the mosquitoes are found, and limited outbreaks have occurred from imported cases in travelers (Los Angeles, Houston, and Phoenix) [31]; and (2) those where the mosquitoes are found, and autochthonous (locally transmitted) outbreaks have occurred (Miami, Brownsville) [31, 32]. The impact of rising temperatures on the risk of dengue outbreaks in the southern United States is an open question.

To explore this question, we adapt the mathematical models developed in [7, 33] to include temperature dependence of the EIP and mosquito lifespan parameters. Our model is composed of ordinary differential equations that describe the time evolution of an outbreak of dengue between humans and adult female mosquitoes. We consider a range of scenarios characterized by different temperatures and initial conditions to understand how temperature affects both dengue risk and mitigation. We compute four *quantities of interest* (QOIs) to evaluate disease invasion potential, transience and severity of outbreaks: (1) the basic reproduction number; (2) the final epidemic size; (3) the timing of the epidemic peak; and (4) the magnitude of the epidemic peak. We compare and evaluate not only the impact of temperature-dependent mosquito lifespan and EIP, but also how the uncertainty in these values impacts disease dynamics and global parameter sensitivity. It then becomes possible to provide a spectrum of possible disease outcomes under a range of climate change scenarios.

## Materials and methods

### Model development

We model the time evolution of a single dengue outbreak in a location without endemic dengue transmission using a compartmental model that tracks human and mosquito populations. The model assumes that the human population is large enough such that mosquitoes are not limited by blood meal availability and the outbreak period is short enough that the total human population remains constant.

The model (Fig 1) divides the total human population, *N*_{h}, into four classes: susceptible *S*_{h}, exposed *E*_{h}, infectious *I*_{h} and recovered *R*_{h}. The susceptible humans, *S*_{h}, move to the exposed class when the human become infected by the bite of an infectious mosquito. Exposed humans, *E*_{h}, move to the infectious class, *I*_{h}, in which humans can infect mosquitoes. Infectious humans, *I*_{h}, recover and move to the recovered class, *R*_{h}. Importantly, our exposed class is comprised of the persons who were infected but not yet infectious. We model the exposed period (latent period) with the intrinsic incubation period (IIP), which is often inferred from the time between a human being infected to having detectable virus [14, 34]. Given that our model is only applicable to a single outbreak, we assume that recovered humans acquire immunity to the dengue virus for the given season [35, 36].

Compartmental diagram for dengue virus transmission and subsequent disease progression across humans and mosquitoes. Susceptible human *S*_{h} become exposed to dengue, *E*_{h}, by the bite of an infectious mosquito *I*_{v}. Exposed persons become infectious, *I*_{h}, after an incubation period, after which they are removed, *R*_{h}, once the infection is cleared. Susceptible mosquitoes, *S*_{v}, enter the system at the rate *h*_{v}(*N*_{v}) where they either die or become exposed, *E*_{v}, when they bite an infected person. Exposed mosquitoes become infectious, *I*_{v}, after the EIP and leave the system though natural death.

The total mosquito population, *N*_{v}, is divided in three classes: susceptible *S*_{v}, exposed *E*_{v}, and infectious *I*_{v}. We assume that mosquitoes enter the susceptible class through birth and that all female mosquitoes are born susceptible. Then susceptible mosquitoes move to the exposed class *E*_{v} via biting a dengue infected human. Exposed mosquitoes move to the infectious class *I*_{v} after completing their extrinsic incubation period, where they remain for the rest of their natural life. We assume that dengue infection does not affect the lifespan of a mosquito. The demographics of the mosquito population follow a logistic growth pattern, i.e. the mosquito birth rate decreases linearly as the populations’ size approach a constant carrying capacity *K*_{v} controlled by the availability of egg-laying sites and competition between larvae [7].

Following the flow diagram given in Fig 1 and the modeling assumptions described above yields the system of equations:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
The total population sizes are *N*_{h} = *S*_{h} + *E*_{h} + *I*_{h} + *R*_{h} and *N*_{v} = *S*_{v} + *E*_{v} + *I*_{v}. The mosquito birth rate is [7]:
(8)
where Ψ_{v} is the natural birth rate in the absence of density dependence, *r*_{v} = Ψ_{v} − *μ*_{v} is the intrinsic growth rate of mosquitoes in the absence of density dependence, and *K*_{v} is the carrying capacity of mosquitoes in the region considered.

The forces of infection from mosquito to humans, λ_{h}(*t*), and from human to mosquitoes, λ_{v}(*t*), are [33, 37]:
(9)
under the assumption that mosquitoes are not limited by available blood meals. *a*_{v} is the constant human-biting rate of the mosquitoes, which is defined as the average number of bites to humans by each mosquito per unit time; *β*_{hv} is the infectivity of an infectious mosquito, which is defined as the probability of pathogen transmission from an infectious mosquito to a susceptible human given that a contact between the two occurs; *β*_{vh} is the infectivity of an infectious human, which is defined as the probability of pathogen transmission from an infectious human to a susceptible mosquito given that a contact between the two occurs; *ν*_{h} is the average rate of time progression of humans from the exposed state to the infectious state; *γ*_{h} is the average human recovery rate; *ν*_{v} is the average rate of time progression of mosquitoes from the exposed state to the infectious state; and *μ*_{v} is the mosquito per-capita natural death rate. Table 1 summarizes the model parameter descriptions and their units.

### Temperature-dependent parameters

We used previous models and data reported from the literature to estimate the temperature dependence in the EIP and lifespan parameters for *A. aegypti* mosquitoes. The temperature-dependent parameter values and ranges for the EIP (1/*ν*_{v}) were taken from the Bayesian lognormal time to event model fit by Chan and Johansson 2012 [14]. The mean value at each temperature is the mean of the posterior distribution and the 95% interval is the middle 95% of the posterior distribution as presented in Tables 2 and 3. These values were explicitly provided for 25°C and 30°C. For 20°C, 35°C, and the temperatures of the three United States cities of interest, these values were estimated from figures. For mosquito lifespan (1/*μ*_{v}), we compiled temperature lifespan data on *A. aegypti* from various studies at temperatures greater than or equal to 15°C [38–40]. We then fit generalized linear models to the data, using the four distributions explored by [14], namely, lognormal, gamma, Weibull, and exponential (see S1 Table for more details). We explored explanatory variables up to degree five polynomials of temperature for all models; however, we used a degree three temperature polynomial for all model types, based on Likelihood Ratio Tests between nested models. All four model types fit the data to a similar extent: a different model performed the best for each of Akaike Information Criteria (AIC) and mean squared error (MSE), and the AIC values were within 20 units of each other. Therefore, we selected the Gamma regression fit for simplicity. We used the prediction from the Gamma regression at a given temperature as the mean parameter value. We then calculated a 95% prediction interval of the modeled mosquito lifespan at every temperature of interest using the method developed by [41]. We summarized these values in Tables 2 and 3 at each temperature of interest.

The 2021 mean temperatures for Los Angeles, Houston, Miami, Brownsville, and Phoenix over June-October were 20.6°C, 27.7°C, 28.4°C, 28.9°C, and 31.8°C respectively (Table 3). Monthly temperatures were obtained from [42].

### Stability analysis

We mathematically analyzed Model (1)–(7) to define the conditions under which the model describes feasible dengue outbreaks and to find the equilibria of the system and their corresponding stability properties. Each equilibrium provides a possible outcome of the dengue outbreak and its corresponding stability properties define the conditions under which a particular result occurs.

In the S1 Appendix, we proved that Model (1)–(7) is epidemiologically and mathematically well-defined in the domain and that for any initial condition in there is a unique solution of System (1)–(7) which remains in for all time *t* > 0. Setting the right-hand sides of the Eqs (1)–(7) equal to zero yields the two equilibria:
where and are any two real numbers. The two equilibrium points have biological meaning when and are in , i.e., , and . The two points and are both disease-free equilibria, i.e., , , and . However, in the steady state the mosquito population dies out ; while in the steady state the mosquitoes achieve their maximal population size *K*_{v}. The following theorems provide the corresponding stability properties of each steady state. We provide their proofs in the S1 Appendix.

**Theorem 1**. *Suppose that* *and* . *Then* *exists for all the parameter values and* *is unstable*.

**Theorem 2**. *Suppose that* *and* . *Then* *exists for all the parameter values and* *belongs to the set X*_{s} = {(*H*_{0} − *R*_{h}, 0, 0, *R*_{h}, *K*_{v}, 0, 0): 0 ≤ *R*_{h} ≤ *H*_{0}}, *which is a local attractor set of the solution set given by System* (1)–(7) *when* . *is unstable when* , *where* (10)

The threshold value is known as the basic reproduction number. The is often interpreted as the expected number of human to mosquito or mosquito to human secondary cases [7, 43]. Here, characterizes the stability of the disease free equilibrium which is unstable and an outbreak occurs when . In this case, the infective curve *I*_{h}(*t*) first increases from an initial *I*_{h}(0) near zero, reaches a peak, and then decreases toward zero as a function of time [44]. As time goes to infinity, the final epidemic size is which satisfies .

We expressed in three terms. The first term, *ν*_{v}/(*μ*_{v} + *ν*_{v}), is the probability that an exposed mosquito will survive the extrinsic incubation period. The second term, *a*_{v}*β*_{vh}*K*_{v}/(*γ*_{h}*H*_{0}), is the average number of newly infections that result from human to mosquito. The third term, , is the average newly infections that result from mosquito to human.

### Sensitivity analysis

We employed two sensitivity analysis methods: Latin Hypercube Sampling (LHS) and the extended Fourier Amplitude Sensitivity Test (eFAST), to quantify the relative impact of model parameter uncertainties on the *quantities of interest* (QOIs): basic reproduction number (, epidemic peak and final epidemic size. All sensitivity analysis was performed through numerical simulations at 20°, 25°, 30°C, and 35° temperature scenarios, from day 0 to 1500 (time relatively large that numerically the disease die out at and beyond of it).

For the Latin Hypercube Sampling [45], we used a uniform distribution to generate 10,000 sampled sets from the parameter space of EIP (1/*ν*_{v}) and mosquito lifespan (1/*μ*_{v}) (Table 2). From these sampled sets, we computed the basic reproduction number () across these parameters and explored at what temperatures the relationship between EIP (1/*ν*_{v}) and mosquito lifespan (1/*μ*_{v}) parameters yielded non-epidemic scenarios ().

For the eFAST method [46], we generated 1,000 samples for each parameter and computed the sensitivity indices of the epidemic peak and final epidemic size. We varied all the parameters in Table 1 simultaneously within the range of values as presented in Tables 2 and 4. We excluded the parameter *H*_{0} in our analysis, despite its impact on the vector-to-host and total susceptible ratios and therefore our QOIs. The qualitative behaviors of the sensitivity indices are both similar when varying or fixing *H*_{0}, as we assumed that *K*_{v} increases proportionally with *H*_{0} and both parameters have the same order of magnitude (the numerical results are not presented in this work). Additionally, we excluded *H*_{0} because we aimed to explore the variability of the QOI to parameters that could be altered through interventions, such as mosquito repellent and reducing vector populations, in order to mitigate disease spread. The resulting magnitudes of the sensitivity indices from the eFAST determine the importance of the parameters on the QOI variability [46, 47]. Specifically, eFAST computes the first order and total order sensitivity indices. The first order sensitivity index represents the fractional contribution of a single parameter to the QOI variance, while the total order sensitivity index takes into account parameter interactions. The difference between the total order and first order is the interaction order sensitivity index, which represents the sole impact of non-linear interactions between parameters on QOI variance. The higher the sensitivity indices are for a given parameter, the larger the influence that parameter exerts on the variance of the QOI. We primarily examined the first order and interaction sensitivity indices for our QOIs, and presented results through visualizations of these metrics.

### Numerical simulations of single outbreaks

We used our model to numerically simulate summer dengue outbreaks at the current mean temperature and at a 3°C temperature increase over June-October for Los Angeles, Houston, Miami, Brownsville, and Phoenix (Table 3). To evaluate the effects of temperature on the risk of dengue outbreaks for these cities, we computed the four QOIs for each city’s simulated outbreaks. Note that to capture the seasonality of the mosquitoes and obtain realistic final epidemic sizes we set the mosquito natural birth rate (*ψ*_{v}) to zero after 180 days into the simulation. Each QOI was computed from simulated outbreaks using the mean values for all parameters. To present the range of possible QOIs in each United States city, we also calculated the QOIs at combined maximum and minimum values of EIP and mosquito lifespan from the 95% intervals, with all other parameters at their mean values. Namely, we examined the following combinations: the largest EIP and smallest lifespan (lowest possible disease spread) and the smallest EIP and largest lifespan (highest possible disease spread) (Table 3) for a particular mean temperature.

In all numerical simulations and sensitivity analyses, we used the parameter values in Table 4 and all initial conditions are zero except *I*_{h}(0) = 1, *S*_{h}(0) = *H*_{0} − 1, and *S*_{v}(0) = *K*_{v}, with *H*_{0} = 1 × 10^{5} for the sensitivity analysis and *H*_{0} being approximations for the 2020 populations for the five United States cities (Los Angeles: 13,109,903; Houston: 7,154,478; Miami: 6,138,333; Brownsville: 421,017; Phoenix: 5,059,909) [51].

## Results

### Impact of temperature-dependent parameters on basic reproduction number

Fig 2 shows the value of as a function of the range of values of EIP (1/*ν*_{v}) and mosquito lifespan (1/*μ*_{v}) consistent with a given temperature. Because is the threshold criteria for an outbreak to occur, the overall inverted ‘U’ shaped effect of temperature on the probability of an outbreak is clear. At low temperatures (20°C), much of the parameter range is not consistent with an outbreak; that is, at this temperature, our theory predicts relatively few outbreaks of small size. However, an increase of 10°C leads to a nearly guaranteed outbreak scenario if dengue is present. Between 30°C and 35°C, the risk of an outbreak plummets to nearly zero due to the fact that almost all of the infected mosquitoes die before they become infectious. Overall, at all temperatures, mosquito lifespan has a stronger effect on than EIP.

This figure shows the basic reproduction number () for the possible values of EIP (1/*ν*_{v}) and mosquito lifespan (1/*μ*_{v}) at 20°C, 25°C, 30°C, and 35°C. Note that since possible ranges of these parameter values differ by temperature (Table 3), the horizontal and vertical axes vary by panel. At 35°C, there is the largest proportion of the parameter space that yields non-outbreak scenarios. At both 25°C and 30°C, the majority of possible EIP and mosquito lifespan values generate such that an outbreak occurs.

### Sensitivity of the epidemic peak and final outbreak size to parametric variance at different temperatures

Figs 3 and 4 show the sensitivity of the peak and final epidemic size to each parameter at 30°C. Both QOIs show very similar patterns with the mosquito biting rate having the largest influence on both QOIs; 50–60% of variance in the QOIs is explained the variance in the biting rate and the interaction between biting rate and other parameters. One key difference in the sensitivity of the QOIs is that the peak epidemic size is much more sensitive to the human infectious period than is the final epidemic size. That is, interventions designed to shorten the period that infected humans are contagious are more effective at reducing the peak size of the epidemic, but less effective at reducing the overall risk. For both QOIs, the interaction of multiple parameters generally explains more of the variance than do the direct effects of most parameters, highlighting the non-linear nature of dengue transmission.

This figure shows the the first and interaction order sensitivity (horizontal axis) of epidemic peak to each parameter (panel). The peak of infectious humans is most sensitive to changes in the biting rate (*a*_{v}), mosquito carrying capacity (*K*_{v}), and human infectious period (*γ*_{h}).

This figure shows the the first and interaction order sensitivity (horizontal axis) of final epidemic size to each parameter (panel). The final epidemic size is most sensitive to changes in the biting rate (*a*_{v}), mosquito carrying capacity (*K*_{v}), and mosquito lifespan (1/*μ*_{v}).

Temperature has a large effect on the sensitivity of the QOIs to parametric variance (Fig 5). As expected, the parameters with direct temperature variability display differential effects on the QOIs at different temperatures. Consistent with the analysis, the role of the mosquito lifespan on the QOIs is much higher at temperatures greater than 30°C because the mosquito lifespan drastically decreases at these temperatures. Above 30°C, the infectious mosquitoes may die before they become infectious and therefore there is a low probability of experiencing an outbreak at 35°C. However, temperature also affects the role of parameters that do not directly have temperature-dependent forms. The impacts of the mosquito carrying capacity and human infectious period, which at 30°C are shown to explain a significant proportion of the variance of the peak of infected humans, increase with temperature only up to 30°C. While sensitivity analysis at a fixed temperature of 30°C demonstrates that a very small proportion of the variance in the final epidemic size is due to the human infectious period (Fig 4), exploring across temperatures, we see that the sensitivity of the human infectious period nearly doubles at 35°C. In contrast, the role of the mosquito carrying capacity on final epidemic size tends to decrease as temperature increases.

This figure shows the first and interaction order sensitivity indices of both the peak number of infected humans (A) and final epidemic size (B) to parameters whose sensitivity varied by temperature. Between 20°C and 30°C, the sensitivity of the QOIs to EIP (1/*ν*_{v}) and mosquito lifespan (1/*μ*_{v}) decreases while the sensitivity of the QOIs to human infectious (1/*ν*_{h}) and mosquito carrying capacity (*K*_{v}) tends to increase. First order sensitivity of the final epidemic size to all parameters except carrying capacity tends to be high at 35°C.

Generally, while the first order sensitivity of the QOIs to the EIP and mosquito lifespan decreases between 25°C and 30°C, the first order sensitivity of the QOIs to the human infectious period and mosquito carrying capacity increases. Temperatures between 25°C and 30°C also yield the largest dengue outbreaks (Figs 2 and 7).

### Effect of temperature on potential dengue dynamics in Los Angeles, Houston, Miami, Brownsville, and Phoenix

We examine simulated dengue outbreaks for five United States cities (Los Angeles, Houston, Miami, Brownsville, and Phoenix) at their current (2021) mean temperatures from June to October and at a 3°C increase from the current mean temperature. Under the 3°C increase (from (27.7° to 30.7°C), Houston has the largest epidemic peak increase (Fig 6). Brownsville has the largest potential peak at current mean temperatures. With a 3°C increase, Los Angeles and Houston’s epidemic peaks increase and Miami and Brownsville’s decrease, while an outbreak does not occur in Phoenix. Generally, temperature increases up to around 31° correspond to increased disease spread. However, temperatures higher than 31°C lead to a decrease in disease spread, and by 35°C there is no dengue outbreak. At the current temperatures, all five cities are at risk of a dengue outbreak, however cities with temperatures around 27–31°C tend to have the largest outbreaks (as seen in Houston’s simulated outbreaks).

This figure depicts the possible epidemic trajectory of dengue in five United States cities at the current mean temperature from June-October for each city and at a 3°C temperature increase from the current mean temperature. At current mean temperatures, fastest disease progression and highest numbers of infected humans are observed Houston, Miami, and Brownsville. With a 3°C mean temperature increase, disease spread decreases for Miami and Brownsville, and is eliminated in Phoenix. In all cities, an outbreak occurs at the current mean temperature.

Houston also has a higher simulated epidemic peak, final epidemic size, and basic reproduction number than either Los Angeles or Phoenix (Fig 7). In addition, Houston has the potential for a much larger epidemic peak and final epidemic size than do those cities, as demonstrated by the bars which represent the range of possible values. Los Angeles and Phoenix tend to have similar values for all QOIs except the time of epidemic peak, where Phoenix’s epidemic peak occurs slightly earlier than Los Angeles’. However, Phoenix has the potential for a higher epidemic peak and final epidemic size than does Los Angeles. Of the cities, the simulated outbreak in Houston leads to the highest number of infected individuals.

This figure shows the epidemic peak, final epidemic size, basic reproduction number (), and time of epidemic peak for Los Angeles, Houston, Miami, Brownsville, and Phoenix. Error bars demonstrate the potential variation in the QOIs for each city. Houston has the potential for a much larger epidemic peak and final epidemic size than the other four cities. Houston, Miami, and Brownsville tend to have higher than the other two cities. The QOIs of the simulated outbreaks for Los Angeles and Phoenix tend to take similar values.

## Discussion

We analyzed a standard mosquito-born disease model to quantify the impact of key temperature-dependent transmission parameters to dengue epidemic outbreaks in three United States cities. We found that all five cities explored (Los Angeles, Houston, Miami, Brownsville, Phoenix) are at risk of a dengue outbreak at their current average temperatures, and that Houston specifically has the highest risk of a large disease burden. With increasing temperatures due to climate change, Los Angeles and Houston could potentially see larger dengue outbreaks, while risk may decrease in Miami and Brownsville. The temperatures in Phoenix may become incompatible with a dengue outbreak. Furthermore, temperatures between 25°C and 30°C yield *A. aegypti* EIP and lifespan values that are most conducive to disease spread, while both lower and higher temperatures tend to generate smaller or no outbreaks. Of the explored parameters, changes in biting rate and mosquito carrying capacity have the largest influence on the variation of the number of infected humans. While the sensitivity of the majority of parameters do not vary by temperature, as temperature increases up to 30°C, the sensitivity of the EIP and mosquito lifespan for the epidemic peak and final epidemic size tend to decrease, and the sensitivity of the human infectious period tends to increases.

Previous studies have supported the assertion that a shorter EIP can result in increased transmission [9, 17]. While our results generally agree with this conclusion, we also show that there is a trade off between EIP and mosquito lifespan. Between 25–30°C, the EIP is small and mosquito lifespan is relatively large, and therefore larger and more quickly moving outbreaks occur at these temperatures. However, the EIP decreases as temperature increases, while adult mosquito lifespan also starts decreasing at 30°C. Our model shows that at around 35°C there is a high probability that even if dengue is introduced, there would be no outbreak. Our sensitivity results also show that total interaction order sensitivity is larger than single order for the final epidemic size, and to a lesser extent, the peak. This indicates that temperature impacts on several parameters may serve as a multiplier to risk changes, having a larger impact than would be indicated by individual parameter sensitivity alone.

Our model also indicates where interventions may have the most relative benefit. Overall, our quantities of interest are most sensitive to the biting rate and mosquito carrying capacity. We should then target interventions at these two parameters (repellent to reduce bites, reducing mosquito density). Interestingly, between 25–30°C, sensitivity of model output to EIP and mosquito lifespan decreases, while sensitivity to the human infectious period increases. This indicates that as temperatures increases within this range (the range of highest disease transmission), human interventions (such as quarantine or preventing onward transmission to mosquitoes) may become more important to limiting disease spread. Also, at higher temperatures, the QOIs are much more sensitive to mosquito lifespan, so reducing the mosquito lifespan can provide more impact and push the model into a zone where outbreaks are unlikely. The final epidemic size is more sensitive to the mosquito lifespan while the epidemic peak is more sensitive to the human infectious period.

Based on our model and its assumptions, regions of the southern United States, including Houston and Los Angeles, could be at increased risk of dengue as temperature rises. However, our models also show that some regions, such as Phoenix, may have reduced risk if temperatures become too hot to sustain an outbreak. Of the cities explored, Houston has the largest risk, and generally temperatures between 27–31°C see the largest outbreaks in our analysis. There has already been limited dengue transmission in Houston and Florida [38, 52, 53]. As temperatures increase we can expect most cities in the southern United States to have a higher risk for dengue. However, this is not limitless, and as temperatures increase above 31°C, outbreaks get smaller and at 35°C there are no outbreaks. A caveat to our results is that mosquito population dynamics also depend on water availability for their life cycle, and adult lifespans also depend on humidity. So, for example, desert regions such as Phoenix may have less potential due to low humidity. Additionally, human infrastructure and behavior impacts the size and duration of dengue outbreaks. In regions with screens, air conditioning, good sanitation and water infrastructure, and access to mosquito repellent, the potential for outbreaks is decreased.

There are several limitations to this work. First, we only consider temperature dependence and not other environmental characteristics including humidity, precipitation, and vegetation. We also only consider the impacts of temperature on the extrinsic incubation period and mosquito lifespan, but other parameters have been shown to be temperature dependent, including the mosquito to human ratio, biting rate, fecundity, egg viability, larval density, development rate, and survival [54–56]. The relationships among these biological parameters, as well as their dependencies on temperature and impacts on disease transmission, are understood to be nonlinear, and in some cases, contradictory. For example, warmer temperatures (within limits) tend to increase fecundity and decrease development times, while decreasing EIP, which tends to increase mosquito abundance and therefore disease transmission. However, warmer temperatures (within limits) can decrease adult mosquito lifespans, which can reduce or even close transmission windows [10, 21, 23]. The temperature dependence in our model and simulations is based on mean temperatures, and does not take into account diurnal fluctuations, which have been shown to impact dengue transmission [10, 57]. Additional modeling studies are needed to incorporate these complexities, and to increase the accuracy of scenario-based projections.

Our sensitivity analysis is dependent on the parameter ranges we select which, while based on values from the literature, could be inaccurate. Similarly, we choose from a uniform range across the parameters, while some parameter combinations may be more or less likely than others. In the future, we plan to incorporate humidity and/or precipitation into the models and expand temperature dependence to other parameters. It may also be important in future iterations of this work to incorporate environmental dependent parameters as distributional forms within the model, and to validate against human case counts and mosquito data across a range of temperature and humidity profiles. Our contribution provides a starting place for better capturing the impacts of temperature on multiple parameters and disease dynamics, in addition to static quantities such as the basic reproduction number.

## Supporting information

### S1 Appendix. Mathematical analysis of the stability of the model system.

https://doi.org/10.1371/journal.pclm.0000115.s001

(PDF)

### S1 Fig. Four model fits to mosquito lifespan data.

Data (points) from [38–40]. All models used a cubic polynomial of temperature as the explanatory variables.

https://doi.org/10.1371/journal.pclm.0000115.s002

(PDF)

### S1 Table. Model equations for mosquito lifespan over time.

Data (points) from [38–40]. Models are of the form Link(*y*) = *α*_{0} + *β*_{1}temperature + *β*_{2}temperature^{2} + *β*_{3}temperature^{3}.

https://doi.org/10.1371/journal.pclm.0000115.s003

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