## Abstract

Personal protection (PP) techniques, such as insecticide-treated nets, repellents and medications, include some of the most important and commonest ways used today to protect individuals from vector-borne infectious diseases. In this study, we explore the possibility that a PP intervention with partial coverage may have the counterintuitive effect of increasing disease burden at the population level, by increasing the biting intensity on the unprotected portion of the population. To this end, we have developed a dynamic model which incorporates parameters that describe the potential effects of PP on vector searching and biting behaviour and calculated its basic reproductive rate, *R*_{0}. *R*_{0} is a well-established threshold of disease risk; the higher *R*_{0} is above unity, the stronger the disease onset intensity. When *R*_{0} is below unity, the disease is typically unable to persist. The model analysis revealed that partial coverage with popular PP techniques can realistically lead to a substantial increase in the reproductive number. An increase in *R*_{0} implies an increase in disease burden and difficulties in eradication efforts within certain parameter regimes. Our findings therefore stress the importance of studying vector behavioural patterns in response to PP interventions for future mitigation of vector-borne diseases.

## 1. Introduction

The collective human mortality rate due to vector-borne diseases is estimated to be more than 1.5 million per annum [1]. Important examples include malaria, dengue, leishmaniasis, yellow fever, Lyme disease and the West Nile virus [2–5].

One of the most important measures of the risk of infectious disease outbreak is the basic reproductive rate, *R*_{0}. *R*_{0} measures the average number of secondary infections caused by a single infection in a naive host population. *R*_{0} thus establishes a threshold criteria for disease invasion; a disease has the potential to invade a population if *R*_{0} > 1, and is unable to persist when *R*_{0} < 1 [6–9]. In general, the higher *R*_{0} is, the more difficult it will be to eradicate the disease [6]. In this study, we explore how personal protection (PP) interventions that protect a portion of individuals in the community affect *R*_{0}.

We use PP to refer broadly to interventions that operate at the individual or household level (rather than the community level), whether or not they kill the vectors in addition to their individualistic protection. Examples include the use of medication, personal use of insect repellent and bednets, including insecticide-treated nets (ITN) [10–14]. Residual spraying of households can also be considered as PP by our definition, if application decisions are made at the household level.

In most applied interventions, only a portion of the population receives or uses a suggested treatment. Although PP provides direct benefits to protected individuals, its population-level effects can be complex and may not always be beneficial in cases where its coverage is not complete. For example, the use of insect repellents by only a part of the population (the treated group, TG) can increase the number of bites on untreated individuals (untreated group, UTG), as repelled vectors seek alternative hosts [15]. When bites are concentrated in a fraction of the host population, the disease risk of the entire population, as represented by *R*_{0}, may increase [11,16–22]. The fact that certain treatments which benefit the TG may indirectly harm the UTG is of interest, since it raises ethical concerns that may limit their usage. An even more troubling scenario is when a PP harms the UTG to such a degree that the overall net effect is an increase in disease risk for the entire population. The possible outcomes of a partial treatment coverage given to a population are summarized in figure 1.

In this study, we apply a model that was originally designed to analyse a system with two host species and one vector species [20,23,24] to a single host population divided into two different groups, the TG and the UTG, and develop it to account for possible effects of PP on vector biting patterns. The goal is to analyse how partial PP coverage affects *R*_{0} under different assumptions about the vector behavioural patterns, in particular, its response to ITN, bednets and insect repellents. The analysis of *R*_{0} identified the conditions under which partial PP coverage will either increase or decrease it.

## 2. Model outlines

We explore how the basic reproductive number, *R*_{0}, depends on individual-level treatment with variable coverage rates of the host population. To calculate *R*_{0}, we modified a previously developed modelling framework [20,24] originally designed for multiple host species, to consider different host types—specifically hosts receiving (TG) and not receiving (UTG) PP treatments. We chose this model because it is a relatively simple framework that includes the details we need to explore the effects of PP on *R*_{0}. Our model considers only one host species: in particular, we do not account here for the possibility that vectors divert from biting human hosts to bite domestic animals or other non-human targets.

The model quantifies the dynamics of the susceptible, infected and removed compartments of the TG and the UTG (denoted by the subscripts T and U, respectively). By using the next-generation-operator technique, we calculate the expression of *R*_{0} from the model equations [7] (see the electronic supplementary material for the model description and equations):
2.1

In equation (2.1), *g _{i}* =

*p*/

_{i}q_{i}*δ*is the transmission ability of host group

_{i}*i*(

*i*=

*T*or

*U*), defined as the product of probability of transmission to a vector

*p*, probability of transmission from a vector

_{i}*q*and the infectiousness duration (1/

_{i}*δ*, see table 1 for parameters definitions) [20].

_{i}*V*and

*d*are the density and the death rate of the vector, respectively,

*k*is the vector biting rate (number of bites per vector per unit time) on hosts belonging to group

_{i}*i*, and

*N*is host group

_{i}*i*density. In equation (2.1), we assumed frequency-dependent biting (i.e. the biting rate of an individual vector is independent of the host total population density,

*N*=

*N*

_{T}+

*N*

_{U}, see the electronic supplementary material). In general, the assumption of frequency-dependent biting leads to a decrease in

*R*

_{0}when host population size increases while vector density remains constant, since bites are then distributed on more host individuals, thus reducing the frequency at which individual hosts are bitten [25,26]. Similar expressions of

*R*

_{0}have been obtained in the past for both, metapopulation [16] and multi-host models [20,26]. The current model builds on these by modelling the factors affecting the biting rates

*k*

_{i}, as elaborated below.

We adapt a ‘classical’ saturated (Holling type II) functional response to calculate vector biting rates on each type of host [27–29]. The type II functional response is a well-known model that has been successfully applied to various consumer–resource systems [30], and which provide a natural basis to model the way vector bites are distributed as a function of searching efficiencies, host densities and handling times with respect to each group. Under these assumptions (see the electronic supplementary material), it can be shown that the vector biting rate on host group *i*, *k _{i}* [27,28,30] is given by:
2.2

*a*2.2

*b*2.2

*c*where 2.3In equation (2.2

*a*),

*h*

_{i}_{1}(

*i*=

*T*or

*U*), the post-biting handling time, is the time the vector spends in handling host

*i*after it has been successfully bitten. Biologically,

*h*

_{i}_{1}may include the time the vector needs for (a) resting and digesting the blood meal, (b) egg production, (c) searching for a proper incubation site and (d) laying eggs.

*h*

_{i}_{2}is the pre-biting handling time, that is, the time the vector is occupied by an individual host when flying around and trying to locate a biting site.

*h*in equation (2.2

_{i}*a*) represents the total handling time, that is, the average time the vector needs to spend in handling a single bite from hosts in group

*i*;

*h*is a weighted sum of the time the vector spends in handling a successful bite (

_{i}*h*

_{i}_{1}+

*h*

_{i}_{2}= the pre- and the post-handling times) and associated in unsuccessful attempts (

*h*

_{i}_{2}= the pre-biting handling time, equation (2.2

*a*)).

*β*(equation (2.3)) is the probability that a biting attempt on a host from group

_{i}*i*will be successful. We assumed that the pre-biting time

*h*

_{i}_{2}is exponentially distributed with mean,

*b*, the protection time, which measures the mean time it takes the vector to achieve a successful bite on group

_{i}*i*.

*A*in equation (2.2

_{i}*b*) represents the general searching efficiency, that is the average biting rate the vector has when foraging in a population of unit density (

*N*= 1) neglecting all handling times (i.e. when

_{i}*h*

_{i}_{1}=

*h*

_{i}_{2}= 0), thus,

*a*can be interpreted as the attractiveness of group

_{i}*i*to the vectors.

To model treatments that may kill the vectors, we assume that the killing acts to decrease the vector longevity, or alternatively, increase its death rate, *d* (provided that its life expectancy is exponentially distributed) [31]. As a simple approximation for simulation purposes, we assume that *d* depends bilinearly on the time the vector is exposed to the pesticide, *h*_{T2}, and the TG coverage rate *x*:
2.4where, *d*_{0} is the basal vector death rate without the killing treatment, and *η* is a constant representing the treatment killing efficiency; it is zero when the treatment does not kill the vector and positive when it does. From the above, *N*_{T} = *xN*, *N*_{u} = (1 − *x*)*N*, where *N* is the total host population density (*N* = *N*_{T} + *N*_{U}). With the aid of equations (2.1)–(2.4), *R*_{0} can be written as:
2.5

where *V*(*x*) is the vector population size when a proportion *x* of the population is treated. In cases where the treatment does not kill the vectors, *V*(*x*) is expected to be constant (assuming that the vector population is at equilibrium), when *η* > 0, *V*(*x*) may decrease with *x* (it can be, of course, that a treatment, such as ITN, does not substantially affect the vector population size) [31]. From the model equations (electronic supplementary material, equation S1), the vector equilibrial population is *r*/*d*, where *r* is the vector's fixed birth rate that is assumed to be limited by other factors in the habitat (e.g. incubation sites, etc.). The expression *r*/*d* enables us to calculate the decrease in the vector population size due to the killing effect of ITN via *d* in equation (2.4). We add the factor exp(−*dL*) to equation (2.1) and obtain equation (2.5) to account for the decrease in *R*_{0} due to a possible vector latency period, *L*. This factor represents the probability that an infected vector will survive the fixed latent period, *L*, and becomes infectious [6,31]. Definitions of all model parameters can be found in table 1.

Every PP can be characterized by different sets of the model parameters (e.g. *A*_{T}, *A*_{U}, *h*_{T}, *h*_{U}, etc.; see also table 1 and equation (2.5)). In this research, we will concentrate mainly on the effects of bednets, ITN, and insect repellents used by individuals, on *R*_{0}. The main goal of this study is to explore the conditions under which these PP techniques will reduce *R*_{0} below its value in an otherwise untreated population, i.e. solving the inequality:
2.6where *R*_{0}(*x*) is *R*_{0} when a proportion *x* of the population is treated (equation (2.5)) and *R*_{0}(0) = *R*_{0}(*x* = 0) = *r*exp(−*d*_{0}*L*) *g*_{u}/(*d*_{0}^{2}Nh_{U}* ^{2}*) (equation (2.5)).

## 3. Results

### 3.1. General consideration

In general, the response of *R*_{0} to PP coverage takes two main forms depending on model parameters (figure 2): (i) the net effect of the PP always decreases *R*_{0}, or (ii) PP increases *R*_{0} if the TG proportion is below a certain threshold *x*_{t}, and decreases *R*_{0} thereafter (figure 2). A detailed analysis of equation (2.5), including an expression of the threshold *x*_{t} for *η* = 0 can be found in the electronic supplementary material. When *R*_{0} is near 1, an increase in it will generally result in an increased population morbidity.

Figure 2 demonstrates that for the chosen parameters, *R*_{0} can reach 2.70 for coverage of 80%, exceeding the value for the untreated population by more than 80% (*R*_{0}(*x* = 0) = 1.5). As such an increase can lead to dramatic rise in disease burden, we will explore how common PP techniques affect the model parameters, and consequently *R*_{0}(*x*), thus enabling us to link between different PP and the behaviour of *R*_{0}(*x*) exemplified in figure 2.

### 3.2. The dependency of *R*_{0} on the use of bednets, insecticide-treated nets and insect repellents

Insect repellents may affect the time the vector spends in trying to bite a host from the treated group, once located, *h*_{T2} (the pre-biting time); the vector may spend less time in trying to bite a host with repellent, or alternatively, it may spend more time around a host with repellent in trying to locate an untreated skin area. The repellent also increases the protection time, *b*_{T}, and it may also make the host less attractive from a distance, thus decreasing *a*_{T} (equation (2.2*b*)). Likewise, bednets may affect *h*_{T2}; they can increase or decrease it, depending on whether nets cause the vectors to give up a protected host quickly, or alternatively, cause them to spend more time in trying to find holes or proximal body parts. Nets also increase *b*_{T}, the protection time. If bednets are also impregnated with insecticide (ITN), they also increase *η*, and consequently the killing rate, *d* (equation (2.4)). Figure 3 illustrates how the form of *R*_{0}(*x*) (the dependency of *R*_{0} on coverage rate, *x*, as exemplified in figure 2) varies over the parameter space of *h*_{T2}, *b*_{T} and *η*. To the best of our knowledge, there are insufficient data available to estimate these parameters with accuracy. We therefore use upper bounds of several days for *h*_{T2} and *b*_{T}—the life expectancy and egg production time of many vectors species [32]. We have also set *d*_{0} = 0.088 d^{−1} and *L* = 0.1 d^{−1}, the death rate and the latency period of typical *Anopheles* spp. (malaria vectors) [6,31]. We have chosen a range of *η* between 0 (bed nets without killing effect) and 0.3. When *η* = 0.3, the vector equilibrial population, *r*/*d* (equation (2.5)) decreases by 23%, an upper bound for a population decrease due to ITN according to a field study on *Anopheles albimanus* [33]. Figure 3 is thus intended to point out on general principles, not to provide quantitative information.

Figure 3 shows parameter ranges where partial PP coverage can increase *R*_{0} (yellow area). The border between the green and the yellow areas shows critical values of the pre-biting and protection times (*h*_{T2} and *b*_{T}, respectively). For example, when *η* = 0.03 and the pre-biting time, *h*_{T2} = 0.39 days (figure 3*b*), there exists a critical value of the TG protection time, *b*_{Tc} = 1.7 days, for which PP reduces *R*_{0} at any coverage level if the protection time is less than *b*_{Tc} (green area), but increases *R*_{0} below a threshold coverage level when the protection time is longer than *b*_{Tc} (yellow area). Thus, if it is harder for the vectors to successfully bite treated hosts (i.e. *b*_{T} increases), the disease risk for the entire population may increase when *R*_{0} is near the threshold value of 1. This is because when *b*_{T} is long, vectors are more likely to be diverted from treated to untreated hosts, concentrating more bites *per capita* on the UTG.

Other parameters behave similarly. When pre-biting time *h*_{T2} is large, *R*_{0} decreases for every coverage rate (green zone), and if it is large enough (e.g. *h*_{T2} > *h*_{T2c} = 1.17d, figure 3*a*), this decrease is independent of the value of *b*_{T}. The killing parameter, *η*, has a critical value as well; the higher it is, the wider the parameter range of *b*_{T} and *h*_{T2} under which *R*_{0} decreases for every PP coverage. If a PP intervention has very strong killing ability (i.e. *η* is high), then it can decrease *R*_{0} irrespective of the coverage rate over the whole range of *h*_{T} and *b*_{T} we studied (in the simulations of figure 3, this occurs, for example, when *η* > 11, data not shown).

## 4. Discussion

In this study, we have investigated the effects of partial PP coverage and found that under some circumstances, it is plausible that partial coverage with popular PP techniques used today, such as ITN, bednets and repellents, can lead to substantial increases in the reproductive number, *R*_{0}. This result is similar to the diversity amplification effect which occurs due to vector preference towards specific host species in a two-species community [20]. In both cases, an increase in *R*_{0} occurs when vectors divert from one host group (TG or less preferred host species) to the other (the UTG or the preferred host species).

Previous models have pointed out the potential negative effects that partial bednet coverage can have due to diversion of vectors from the TG to the UTG, when combined with low killing efficiency [11,21,34]. The effects estimated, however, were very low, particularly at the level of the entire population [11,31,32]. It has also been speculated that total population morbidity could increase with increased bednet coverage in a case where bednets were combined with vaccines that could affect the population immunity of specific vulnerable host groups [35]. Field studies, however, praise the use of ITN for their success in reducing malaria incidence, or in decreasing other important metrics of disease risk (e.g. entomological inoculation rate, human biting rate and vector population) [36–41].

This study is the first to systematically explore the effect of partial PP coverage on *R*_{0} over a wide range of plausible parameters, and the first to find a potential for substantial increase in population-level risk; when *R*_{0} is near its threshold (i.e. 1), any increase in it is expected to lead an increased population-level morbidity. It is important to note, however, that when morbidity and force of infection are high, an increase in *R*_{0} caused by protecting part of the population will not always be expected to increase population-level morbidity, since the effects of increasing *R*_{0} would be outweighed by the direct effect of protecting part of the population. Yet, the increase in *R*_{0} in these cases may still lead to an increased risk in a portion of the population.

Our model makes the subtle yet important distinction between protection and diversion [11,21,31]. In previous studies, both diversion and protection have been related to the probability that a vector will give up a protected host and turn to look for another victim [11,21,31]. In the present framework, however, diversion is equivalent to the time duration the vector spends in trying to bite a protected host, *h*_{T2}, that is, its pre-biting time. The shorter the pre-biting time, *h*_{T2}, the stronger the diversion effect of the respective PP. The protection, *b*_{T}, in our model, is equivalent to the mean time the vector needs for a successful biting attempt of a protected host, irrespective of the time the vector actually spends in that attempt (*h*_{T2}). In our study therefore, protection and diversion are two independent properties, and hence may be affected differently by different PP interventions. Figure 3 demonstrates the counterintuitive dependency of *R*_{0} on protection due to this differentiation; a more protective PP can increase *R*_{0} when the vector pre-biting handling time, *h*_{T2}, is short, and the coverage rate is below a certain threshold. For longer *h*_{T2}, however, every PP coverage reduces *R*_{0} (figure 3). Consequently, an increase in *R*_{0} may therefore occur due to a change in the vector foraging behaviour. If vectors switch quickly on encountering nets (or on encountering ITNs), the pre-biting time, *h*_{T2}, will decrease, and consequently *R*_{0} will increase for some levels of coverage (assuming high enough protection times, figure 3). Such a change of the vector behaviour is realistic. Avoiding landing on ITNs and spending less time in trying to bite protected individuals are traits that can increase vector longevity and fitness, and thus may spread within the population relatively fast. Changes in vector behaviour correlated with ITN usage have already been observed: mosquitoes change their activity time, host species (from human to livestock) and feeding site (indoor or outdoor) within several years in high coverage areas [39,42–45]. Unfortunately, to the best of our knowledge, there are no field or laboratory measurements regarding the time allocation used by vectors while foraging for potential hosts.

This study expands on previous work by supplementing simulation-based exploration of the various effects of PP on disease burden with analytical results on *R*_{0}, thus increasing generality and providing a better mechanistic understanding [46]. Our results are most applicable to cases where *R*_{0} is near its threshold (i.e. 1). Under such circumstances, an increase in morbidity may occur for PP partial coverage rates if certain vector behavioural patterns exist, especially a decrease in its pre-biting handling time. This study, therefore, stresses the importance of field research on the vector's time allocation to foraging for potential hosts and its relation to the PP techniques widely used today for future elimination and mitigation of vector-borne infectious diseases.

## Authors' contributions

E.M. and A.H. conceived the study. E.M. wrote the first draft of the manuscript. All authors revised the manuscript, contributed to later drafts, developed the model and analysed the results. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

This study was supported by the Bill and Melinda Gates Foundation Global Health Program http://www.gatesfoundation.org/ (grant no. OPPGH5336). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

- Received July 26, 2015.
- Accepted February 3, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.