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34 References# Approximation of the Basic Reproduction Number R 0 for Vector-Borne Diseases with a Periodic Vector Population

**Abstract**

The main purpose of this paper is to give an approximate formula involving two terms for the basic reproduction number R(0) of a vector-borne disease when the vector population has small seasonal fluctuations of the form p(t) = p(0) (1+epsilon cos(omegat - phi)) with epsilon < 1. The first term is similar to the case of a constant vector population p but with p replaced by the average vector population p(0). The maximum correction due to the second term is (epsilon(2)/8)% and always tends to decrease R(0). The basic reproduction number R(0) is defined through the spectral radius of a linear integral operator. Four numerical methods for the computation of R(0) are compared using as example a model for the 2005/2006 chikungunya epidemic in La Réunion. The approximate formula and the numerical methods can be used for many other epidemic models with seasonality.

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- "Previous work has shown that the potential for disease outbreaks can be influenced by factors such as human population size, vector abundance, seasonality in transmission, and connectivity of populations [63][64][65][66][67][68][69][70]. In this study, we consider a number of scenarios to address the impact of some of these factors on outbreaks of dengue in the Miami UA. "

[Show abstract] [Hide abstract]**ABSTRACT:**Expansion of mosquito-borne pathogens into more temperate regions of the world necessitates tools such as mathematical models for understanding the factors that contribute to the introduction and emergence of a disease in populations naïve to the disease. Often, these models are not developed and analyzed until after a pathogen is detected in a population. In this study, we develop a spatially explicit stochastic model parameterized with publicly available U.S. Census data for studying the potential for disease spread in Urbanized Areas of the United States. To illustrate the utility of the model, we specifically study the potential for introductions of dengue to lead to autochthonous transmission and outbreaks in a population representative of the Miami Urbanized Area, where introductions of dengue have occurred frequently in recent years. We describe seasonal fluctuations in mosquito populations by fitting a population model to trap data provided by the Miami-Dade Mosquito Control Division. We show that the timing and location of introduced cases could play an important role in determining both the probability that local transmission occurs as well as the total number of cases throughout the entire region following introduction. We show that at low rates of clinical presentation, small outbreaks of dengue could go completely undetected during a season, which may confound mitigation efforts that rely upon detection. We discuss the sensitivity of the model to several critical parameter values that are currently poorly characterized and motivate the collection of additional data to strengthen the predictive power of this and similar models. Finally, we emphasize the utility of the general structure of this model in studying mosquito-borne diseases such as chikungunya and Zika virus in other regions.- "The expressions for R 0 in equations [2.5,2.7,2.8] do not apply in seasonal settings, but can be calculated numerically using Floquet theory [35]—see the electronic supplementary material. "

[Show abstract] [Hide abstract]**ABSTRACT:**There is substantial variation in the relapse frequency of Plasmodium vivax malaria, with fast-relapsing strains in tropical areas, and slow-relapsing strains in temperate areas with seasonal transmission. We hypothesize that much of the phenotypic diversity in P. vivax relapses arises from selection of relapse frequency to optimize transmission potential in a given environment, in a process similar to the virulence trade-off hypothesis. We develop mathematical models of P. vivax transmission and calculate the basic reproduction number R0 to investigate how transmission potential varies with relapse frequency and seasonality. In tropical zones with year-round transmission, transmission potential is optimized at intermediate relapse frequencies of two to three months: slower-relapsing strains increase the opportunity for onward transmission to mosquitoes, but also increase the risk of being outcompeted by faster-relapsing strains. Seasonality is an important driver of relapse frequency for temperate strains, with the time to first relapse predicted to be six to nine months, coinciding with the duration between seasonal transmission peaks. We predict that there is a threshold degree of seasonality, below which fast-relapsing tropical strains are selected for, and above which slowrelapsing temperate strains dominate, providing an explanation for the observed global distribution of relapse phenotypes.- "In [1] (see equation (51)), it was shown that, for small b, we have "

[Show abstract] [Hide abstract]**ABSTRACT:**For a family of periodic SEIRS models with general incidence, we prove the existence of at least one endemic periodic orbit when R_0>1. Additionally, we prove the existence of a unique disease-free periodic orbit, that is globally asymptotically stable when R_0<1. In particular, our main result establishes a sharp threshold between existence and non-existence of endemic periodic orbits for this family of models.

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