Models to assess how best to replace dengue virus vectors with Wolbachia-infected mosquito populations

Highlights

• Now various strains of Wolbachia-infected vectors are released in dengue areas.

• Models with and without augmentation for dengue control are proposed.

• Stability analysis shows there are backward bifurcations and multiple attractors.

• Initial values and release methods affect the success of population replacement.

• Suitable Wolbachia and release methods are needed for the control of dengue virus.

Abstract

Dengue fever is increasing in importance in the tropics and subtropics. Endosymbiotic Wolbachia bacteria as novel control methods can reduce the ability of virus transmission. So, many mosquitoes infected with Wolbachia are released in some countries so that strategies for population replacement can be fulfilled. However, not all of these field trails are successful, for example, releases on Tri Nguyen Island, Vietnam in 2013 failed. Thus, we evaluated a series of relevant issues such as (a) why do some releases fail? (b) What affects the success of population replacement? And (c) Whether or not augmentation can block the dengue diseases in field trials. If not, how we can success be achieved? Models with and without augmentation, incorporating the effects of cytoplasmic incompatibility (CI) and fitness effects are proposed to describe the spread of Wolbachia in mosquito populations. Stability analysis revealed that backward bifurcations and multiple attractors may exist, which indicate that initial quantities of infected and uninfected mosquitoes, augmentation methods (timing, quantity, order and frequency) may affect the success of the strategies. The results show that successful population replacement will rely on selection of suitable strains of Wolbachia and careful design of augmentation methods.

Introduction

The mosquito-borne diseases, dengue fever and dengue haemorrhagic fever, are among the leading causes of illness in the tropics and subtropics such as the Americas, Southeast Asia, the Western Pacific, Africa and the Eastern Mediterranean, mostly in urban and semi-urban areas, with up to 380 million infections estimated to occur annually [1]. People may be infected with dengue more than once because it is caused by any one of four related viruses transmitted by mosquitoes, especially Aedes aegypti and Aedes albopictus [2]. There are no antiviral therapies or vaccines available to prevent infection with dengue, so control of mosquitoes remains the most effective protective measure for dengue prevention [3], [4]. However, the traditional use of insecticides as a control measure is often prohibitively expensive and environmental undesirable; moreover, it may lead to insecticide resistance [3].

Therefore, it is necessary to search for novel technologies to break dengue transmission cycles [5], [6]. At present, exploitation of Wolbachia bacteria is a promising method for manipulation of mosquito vectors. Wolbachia are maternally transmitted endosymbiotic bacteria, estimated to infect up to 65% of insect species and approximately 28% of the surveyed mosquito species [7], [8]. The bacteria live within testes and ovaries of their hosts and are passed from one generation to the next through the hosts’ eggs; thus, they can interfere with the insects’ reproductive mechanisms, causing phenomena such as cytoplasmic incompatibility (CI), parthenogenesis and feminization of genetic males. Appearances of these phenotypes depend on the host species and Wolbachia types. CI causes uninfected females that mate with infected males to rarely produce fertile eggs, while infected females are not affected. This gives infected females an advantage and helps the bacteria to spread quickly through a mosquito population [9], [10], [11], [12].

Two strategies to develop Wolbachia for biological control of dengue virus have been proposed [13]. Different strategies lead to the selection of different strains of Wolbachia and methods for augmentation (pulse), which involves the supplemental release of infected mosquitoes. Relatively small seedings may be released at a critical time of the season (inoculative release) or millions may be released (inundative release) when the density of infected individuals is too low [14], [15]. Population suppression (mosquitoes dying out) based on CI occurs if only infected males are inundatively released, as in the case of the successful suppression of Culex pipiens populations in field tests [16], [17]. Some strains of Wolbachia can shorten the mosquitoes’ lives, indirectly preventing viral maturation and transmission [18]. Furthermore others can not only successfully spread within mosquito populations but also act as a ‘vaccine’ for the mosquitoes to stop them from replicating and transmitting many types of viruses including dengue virus [19], [20]. So the strategies of population replacement (ensuring uninfected mosquitoes are replaced by infected ones) based on CI mechanisms and matrilineal inheritance, have been proposed involving inoculative releases of infected mosquitoes [19], [20], [21], [22].

Many mathematical models have been investigated for the spread of Wolbachia infection [23], [28], [29], [30], [31], [32], [33], [34]. A continuous-time model for the behaviour of one and two strains of Wolbachia within a single well-mixed population has been studied which demonstrated the Allee effect and founder control. Patchy persistence of the two strains has been shown in a discrete spatial model [23], [28]. Delay differential equations analyze how the reproductive advantage offsets the fitness costs for the success of population replacement [32]. Moreover, birth-pulse models of Wolbachia-induced CI have studied the effect of different density dependent death rate functions on different strategies for control of dengue virus [34].

At present dengue diseases, being among the leading causes of illness in the tropics and subtropics, have attracted close attention all over the world. After receiving government approval and support from the local community, researchers in many countries aim to release mosquitoes implanted with different strains of Wolbachia bacteria to block the spread of dengue virus. The first releases were of mosquitoes infected with wMel Wolbachia (strong anti-dengue properties and low fitness costs) in Yorkeys Knob and Gordonvale in north-eastern Australia in 2011 [20]. Subsequentially, in Tri Nguyen Island, Vietnam, two types of Wolbachia-infected mosquitoes involving wMelPop (reducing mosquito lifespan) and wMel were released in April 2013, which failed, and in May 2014 (on going), respectively [35]. In communities around Yogyakarta, Indonesia, mosquitoes infected with wMel Wolbachia were also released in January 2014.

At present, over the next 30 years 7 million dengue cases are reported in Brazil. Today the country leads the world in the number of dengue cases with 3.2 million cases and 800 deaths reported during 2009–2014 [36]. Ten thousand mosquitoes will be released there once a week for three to four months. The first release was in September 2014 in Tubiacanga, in the north of Rio de Janeiro, to block the spread of dengue virus. Three more neighbourhoods will be targeted next, and large scale studies to evaluate the effect of the strategy are planned for 2016. In addition, further trials are also planned for Colombia [36]. It is useful to generalize analysis of the strategies for possible application in other countries with high-prevalence areas of dengue diseases. For instance, countries such as Malaysia, Singapore and China reported more cases in 2014 when compared to the same period in 2013 [37].

However, not all of field trails are successful in different countries, thus, interesting issues arise including (a) why did some releases fail in the end? (b) What affects the success of population replacement? (c) Whether or not augmentation can block dengue diseases in field trials, such as in Brazil? If not, how we can it be done successfully? In this study, we focus on answering these questions through mathematical models. Firstly, a continuous four-dimensional mosquito model with Wolbachia-induced CI is proposed and simplified as a two-dimensional model, because the ratio of males to females in each state is assumed to be identical. Secondly, we analyze the existence and stability of equilibria, and the conditions of backward bifurcations in three cases. The results show that the zero equilibrium is always unstable, which indicates population eradication will not be achieved, so only population replacement will be considered. If forward bifurcation occurs, the condition of the threshold R₀ > 1 ensures the success of population replacement naturally. However, as the threshold value R₀ may be very small when the mosquito population begins to become infected with Wolbachia, it is unlikely for R₀ to be larger than one in practice. Therefore it is weakened by the existence of backward bifurcation. When the fitness cost is large enough, there exists a backward bifurcation which is very important in disease control. So we discuss the basins of attraction of the two attractors and analyze how the parameter space impacts on the success of control strategies without augmentation. Thirdly, regarding the release of infected mosquitoes, models with finite and infinite augmentation are considered to analyze the effects of the initial densities of mosquitoes, augmentation timings, augmentation quantities and numbers of augmentations on the success of population replacement in the general case. The results show that suitable strains of Wolbachia should be selected and augmentation methods should be carefully designed for successful population replacement.