Impact of Behavioural Changes in Mosquito Feeding on Malaria Invasion: A Model-Based Approach

Background: Malaria hosts are known to manipulate the feeding behaviour of mosquitoes to protect them from external threats and control. In particular, the phenomenon in which a mosquito’s feeding target is biased toward an infectious host is called a vector-bias, and it can be a threat to malaria eradication if not considered. Aim of the study is to understand the problems that may arise when vector-bias is not considered in early invasion scenarios. Methods: Stochastic formulations of malaria transmission, including the vector-bias effect, ware constructed. Invasive dynamics were investigated using an individual-based continuous time Markov chain model and the offspring distributions of secondary infections. In addition, the extinction probability was derived using the negative binomial count model. Results: Invasions will occur quickly, and once the disease spreads, extinction will become difficult compared to when the vector-bias effect is not considered. In a highly heterogeneous environment, vector-bias has rare effect on decreasing the extinction probability. Conclusion: The early detection of a malaria invasion and the early control beginning are more important due to vector-bias for the malaria eradication in early invasion scenarios. In addition, some possible mosquito-biased behaviours were discussed in terms of adaptive dynamics.


Introduction
The adaptive host manipulation hypothesis [1] states that parasites evolve themselves to control specific aspects of their host's behaviour and thereby enhance Full list of author information is available at the end of the article attributes that are exploited by parasites to attract mosquitos [3,4,5,6,7]. During the 1980s, an experiment was conducted using mice [8,9], hens [10], and lambs [11], in which mosquitoes were found to feed preferentially on infected hosts. A decade earlier, a similar result was found in the case of malaria in a human host [12], in which it was determined that malaria-infected humans have a greater attractiveness to mosquitoes, a phenomenon that was later called vector-bias [13], and more evidences of mosquito's feeding bias toward Plasmodium-infected vertebrates were found [14,15,16,17,18,19]. Such studies encouraged thinking over whether the parasites manipulate their hosts to increase their probability of survival or not.
Such manipulation of mosquitoes by malaria parasites cause significant difficulties in predicting or controlling malaria [20,21]. Therefore, the mosquito feeding behaviour is important in epidemiological studies of malaria transmission. Numerous modelling attempts have been made to analyse or quantify the effects of mosquitoes on the changes in feeding behaviour [13,22,23,24]. The first mathematical model, which dealt with mosquito behaviour increasing the total biting rate (the total biting rate of susceptible and infected mosquitoes), proved to increase the equilibrium infection level during endemic equilibrium, whereas infected host preference was found to exert both increasing and decreasing effects [22]. Discussions regarding impact of the manipulation of mosquitoes by malaria parasites to the transmission probability of malaria diseases was presented by Cator et al. [25,26]. In [13], the authors presented a vector-bias model and concluded that mosquitoes are most preferred by a high prevalence of the parasites. Although both positive and negative effects were reported quantitatively, no argument for this dual role has been presented, but was recently clarified in [27]. It was shown that mosquito encounters with susceptible hosts are as likely as mosquito encounters with infected hosts, but that bites occur in infected hosts with a higher probability. Thereafter, numerous studies on vector-bias were conducted, with a focus on the periodicity of mosquito abundance [28], time-delay [29], fractional-order [30], spatial structure [31], and real-world application [32]. It can be seen that the aforementioned studies dealing with the modelling of vector-bias focused on the impacts of the long-term behavioural dynamics.
However, there have been few studies on how vectorbias provides momentum for the spread of the disease during the initial stages of malaria transmission. Motivated by this, in this paper, we investigated the impact of vector-bias at the initial stages of Plasmodium falciparum malaria transmission using a stochastic method, by comparing the dynamics when considering vectorbias and when not considering it. Mainly, malaria invasion and extinction were studied. Since early dynamics such as disease invasion are known to be better understood in stochastic models than in deterministic models [33,34], and are also sensitive to changes in the behaviour of each individual, an individual-based continuous time Markov chain model of the underlying approach introduced in this study was constructed.
The suggested formulation allows applying Gillespie's stochastic simulation when considering the biased behaviour of each mosquito [35,36,37]. To consider the transmission heterogeneity and consequent extinction probabilities, the offspring distribution along the negative binomial count model was calculated. Finally, the impact of vector-bias on malaria control was discussed.
We consider the birth and death rates to be the same for both the host and the mosquito, as symbolized by µ h for the host and µ v for the mosquito. In addition, ξ is the per capita rate of the loss of immunity of the recovered host, and p h and p v are the probability of mosquito-to-human and human-to-mosquito transmission of a disease, respectively, for each bite. λ is called a bias parameter. Notably, λ = 1 means that there is no vector-bias and the vector-bias increases as the value of λ > 1 increases. The total human population The basic reproduction number, which is the average number of secondary infected hosts generated by a single infected host introduced under a wholly susceptible state [38], is given by the following: should be noted that λ increases the basic reproduction number and enforces a breakout.

Distribution of a bias parameter
The estimated parameter λ were fitted to a Weibull distribution with the probability density function where the scale parameter α = 4.34 and the shape parameter β = 0.83 (i.e., the expected value of λ,λ, is 4.78 and the coefficient of variation is 0.58) in [27] based on the assay conducted in [12]. In Figure 1, the suitability of a Weibull distribution fitting of the bias parameter was demonstrated (see Figure 1 in [27]).
where t ∈ [0, ∞). We denote as the transition probability associated with the stochastic process for ∆t > 0, not fixed but the duration at which something happens next, which is defined as follows:   Gillespie's algorithm is that it first determines when the next event will happen. Suppose that the current time is t. Second, it determines the event that is next to occur by randomly applying the probability following the processing rates. As we showed in the previous section, the experimental results indicate that the biased behaviours of each mosquito differ significantly.
Following this, we consider a refined model ( bias parameter values to be assigned to each mosquito.
From these procedures, we can divide the two events of human infection and mosquito infection, into i v independent sub-events of human infection and s v independent sub-events of mosquito infection. This model is called as individual-based CTMC model. Table 1 shows a summary of the events of the individual-based CTMC model. A detailed description of the simulation procedure of the model is shown in Figure 2.
Because λ is fitted with a Weibull distribution, the randomly selected bias parameter is given as for the random number r 3 , following a uniform distribution, Unif(0, 1). This iteration is repeated until the final limit is reached or there is no reaction in the system.
Branching process formulation of the underlying model When the outbreak is limited, it is assumed that one infection occurs in a wholly susceptible human population, Z 0 = 1, and that the offspring distribution of each person in each generation is an independent and identically distributed random variable, Z 1 . Applying q k = P(Z 1 = k), the extinction probability is then defined as the smallest fixed point of G Z1 , that is, the extinction probability is the smallest solution within where G Z1 is the probability generating function of q k .
To show the effect of the vector-bias in the extinction of malaria, we calculated the extinction probability using the varying bias parameter λ and dispersion parameter s. In many cases, it is difficult to find the solution of (4) analytically. Hence, we used a numeri-

Results
In this section, we investigated the impact of vectorbias on malaria invasion and extinction under a lowtransmission setting. The parameter values adopted in the simulation are listed in Table 2.

Description, Notation (Unit) Value Source
Human birth rate and death rate,

× 10 −5 Estimated
Mosquito birth rate and death rate, µv (day −1 ) 0.10 [27] Biting rate of a mosquito, b (day −1 ) 0.35 [27] Per capita recovery rate, γ 3.50 × 10 −3 [27] Per capita rate of loss of immunity, ξ (day −1 ) 2.74 × 10 −3 [27] Transmission probability of infection from an infectious mosquito to a non-malaria infected human when a contact between the two occurs, p h (1) 0.024 [27] Transmission probability of infection from an infectious human to a susceptible mosquito when a contact between the two occurs, pv (1) 0.24 [27] I h = k as  supports previous the results in [27], which show that the effect lessens the endemicity in a high transmission area and increases it in a low transmission area.

Invasion of malaria
We In addition, R h (0) = 0 was assumed. Table 3 shows the probability that an infection will occur, and the probability of a secondary infection of 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64   mosquitos and humans, as two invasion scenarios [46] and the time taken for the secondary infection of a human to occur, under the human invasion scenario. In both cases, the probability of infection is doubled when considering the vector-bias. When an infection occurs, the probability that malaria will persist increases due to the vector-bias. This is because the number of infected people increases after the first invader arrives, whether a mosquito or human. When there is an invasion by one person, vector-bias has a three-fold impact on the results. Therefore, it is important to apply treatment prior to the invasion because the probability of persistence increases rapidly, once the initial invasion occurs. based on the expression for R 0 , we know that an increasing λ increases R 0 . Therefore, an increase in λ will also reduce the disease extinction probability. Thus, it can be seen that the vector-bias reduces the chance of malaria extinction in some endemic or invasive communities. Although vector-bias lessens the endemic level in a high transmission area [27], it provides a positive potential to maintain the malaria endemic.
The right panel of Figure 6 shows that the vectorbias has no effect on reducing the malaria extinction probability in a highly heterogeneous environment. Although the bias towards infectious humans is 20-times higher, the extinction probability is only reduced to 0.23% (0.9993 to 0.9970). However, as heterogeneity decreases, the decreasing rate of extinction probability owing to vector-bias increases. It can be seen that   the extinction probability decreases rapidly as the bias parameter increases but remains above a certain level.
In terms of the adaptive host manipulation hypothesis in biology, these results suggest two possibilities: first, an extreme-biased behaviour of mosquitos can occur in areas having a high heterogeneity to increase the viability of a malaria parasite, and second, a low level of biased behaviour is sufficient to increase the viability of the parasite in areas with a low heterogeneity. probability. This is because the vector-bias accelerates the initial invasion speed owing to an increase in R 0 , which makes the extinction more difficult, by rapidly making the endemicity exceed the extinction threshold and increasing the extinction probability.