# Modelling the effects of malaria infection on mosquito biting behaviour and attractiveness of humans

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## Abstract

We develop and analyse a deterministic population-based ordinary differential equation of malaria transmission to consider the impact of three common assumptions of malaria models: (1) malaria infection does not change the attractiveness of humans to mosquitoes; (2) exposed mosquitoes (infected with malaria but not yet infectious to humans) have the same biting rate as susceptible mosquitoes; and (3) mosquitoes infectious to humans have the same biting rate as susceptible mosquitoes. We calculate the basic reproductive number, \(R_0\), for this model and show the existence of a transcritical bifurcation at \(R_0=1\), in common with most epidemiological models. We further show that for some sets of parameter values, this bifurcation can be backward (subcritical). We show with numerical simulations that increasing the relative attractiveness of infectious humans, increases \(R_0\) but reduces the equilibrium prevalence of infectious humans; decreasing the biting rate of exposed mosquitoes increases \(R_0\) and the equilibrium prevalence of infectious humans and mosquitoes; and increasing the biting rate of infectious mosquitoes has no impact on \(R_0\) or the equilibrium prevalence of infectious humans, but decreases the infectious prevalence of infectious mosquitoes. These analyses of a simple malaria model show that common assumptions around the relative attractiveness of infectious humans and the relative biting rates of exposed and infectious mosquitoes can have substantial and counter-intuitive effects on malaria transmission dynamics.

## Keywords

Mathematical model Malaria Bifurcation analysis Mosquito biting## Mathematics Subject Classification

92D30 34C23## 1 Introduction

Malaria is an infectious disease of humans, usually transmitted through the bites of Anopheline mosquitoes. Female mosquitoes bite humans (and other warm-blooded animals) for blood meals to provide protein for egg development. During each feed, the mosquito injects some saliva into the host to prevent clotting before sucking the blood into its stomach. After ingesting the blood, the mosquito rests for two to three days (depending on the ambient temperature) while it digests the blood. The mosquito then searches for a water body where it lays the eggs and then seeks another host to repeat its feeding cycle [23].

Mosquitoes experience different levels of mortality risk during the different phases of the feeding cycle. Host-seeking and attempting to feed on hosts is the most dangerous part of the cycle, where mosquitoes are most likely to die. The resting phase is usually the safest where mosquitoes are stationary. However, the mortality risks of the different stages can be altered by malaria control interventions. For example, insecticide-treated nets increase the mortality of host-seeking mosquitoes and indoor residual spraying with insecticides increases the mortality of resting mosquitoes [12].

Infectious mosquitoes contain malaria sporozoites in their salivary glands and infect humans when they inject saliva into them. In humans, the sporozoites pass through liver and asexual blood stages before transforming into infective sexual stages (called gametocytes). The incubation period in humans is relatively short (about 20 days) but the infectious period can last for multiple months. Humans infect mosquitoes when a mosquito sucks up a male and a female gametocyte during her blood meal. The gametocytes then fuse in the mosquito’s stomach to eventually form an oocyst that releases sporozoites (that then travel to the mosquito’s salivary glands to complete the malaria life cycle). The incubation period in the mosquito is about 10–12 days long, which is on the same order as the mosquito’s life span so it has a substantial effect on malaria epidemiology and control [16].

Recent evidence has shown that malaria parasites can affect the mosquito’s host-seeking behaviour to increase their probability of transmission [1, 19, 20]. Mosquitoes with oocysts (who are infected but not yet infectious) tend to be less mobile and spend more time resting. They therefore have longer feeding cycles and experience a lower mortality rate than uninfected mosquitoes. Mosquitoes with sporozoites tend to be more restless and are likely to take multiple blood meals in a single feeding cycle (that is, they bite multiple hosts before resting). Therefore, although they experience a higher mortality rate than uninfected mosquitoes, they also feed more frequently and thereby infect humans more often than they would if their feeding rate was unchanged.

Additionally, there is evidence that humans with a high density of gametocytes are more attractive to mosquitoes than humans without gametocytes [21], thereby making it easier for infectious humans to transmit to mosquitoes.

There is a long history of mathematical models of malaria starting with Ronald Ross’ first ordinary differential equation (ODE) model for the proportion of infectious humans and mosquitoes [26]. Most models developed since then have been similar deterministic population-based models [2, 3, 10, 24, 25] although recently stochastic individual-based simulation models have increased in prominence [17, 28]. However, these models have rarely considered the impact of malaria infection in mosquitoes on their feeding frequency or the impact of malaria infection in humans on their attractiveness to mosquitoes.

Here we develop a simple deterministic population-based model that extends existing models to include these effects. We first describe the model and perform a qualitative analysis to show the existence of disease-free and endemic equilibrium points, and the possibility of a backward transcritical bifurcation that is common to many such malaria models. We finally perform some numerical simulations that illustrate the effects of varying the infection-state dependent mosquito biting frequency and the relative attractiveness of infectious humans.

## 2 The model

We assume that the mosquito population is divided into three disjoint compartments: susceptible, exposed (mosquitoes carrying oocysts), and infectious (mosquitoes carrying malaria sporozoites). We denote the three populations at time *t* as \(S_v(t)\), \(E_v(t)\) and \(I_v(t)\), respectively. In order to account for the effects of malaria parasites on the mosquito’s host-seeking behaviour [1, 19, 20], we introduce two positive constants, \(\varphi _E\), and \(\varphi _I\), to represent the biting rate of exposed and infectious mosquitoes relative to susceptible mosquitoes, with \(\varphi _E <1\) and \(\varphi _I >1\).

*vector-bias*parameter) by

*p*. For simplicity, we assume that humans infected with malaria are always infectious; thus ignoring the incubation period and any variations in gametocyte density over time.

Description and baseline values/range of parameters of model (1)

Description | Baseline value/range | Source | |
---|---|---|---|

\(\Lambda _h\) | Human birth rate | (1000/65) humans/year | – |

\(\mu \) | Human death rate | 1/65 years\(^{-1}\) | [11] |

\(\alpha \) | Human malaria death rate | \(9\times 10^{-5}\) days\(^{-1}\) | [11] |

\(\delta \) | Human recovery rate | 1/211.6 days\(^{-1}\) | [27] |

\(\Lambda _v\) | Mosquito emergence rate | \((10^{4}/9)\) mosquitoes/day | – |

\(\eta \) | Base mosquito death rate | 1/9 days\(^{-1}\) | [22] |

\(\omega \) | Base mosquito biting rate | 1/2 days\(^{-1}\) | [11] |

\(\gamma \) | Mosquito progression rate to infectious class | 0.091 days\(^{-1}\) | [11] |

\(\pi _h\) | Prob. of malaria transmission to humans | 0.022 | [11] |

\(\pi _m\) | Prob. of malaria transmission to mosquitoes | 0.48 | [11] |

\(\varphi _E\) | Relative biting rate of exposed mosquitoes to susceptible mosquitoes | [0, 1] | [8] |

\(\varphi _I\) | Relative biting rate of infectious mosquitoes to susceptible mosquitoes | [1, 5] | [8] |

| Relative attractiveness of infectious humans as compared to susceptible humans | 2 | [21] |

*forces of infection*on humans and vectors, respectively. The infection rate per susceptible human and per susceptible vector are given, respectively, by,

## 3 Local stability analysis of the disease-free equilibrium point

*F*and the ‘transition’ matrix

*V*defined in [14] are given, respectively, by

### **Theorem 1**

The disease-free equilibrium \(E_0\), given by (3), is locally asymptotically stable if \(R_0 <1\) and unstable if \(R_0 >1\), where \(R_0\) is given by (4).

## 4 Endemic equilibrium points

*endemic*, we mean an equilibrium of system (1) where all components are positive. The components are solutions of the following system,

- (1)
There is a unique endemic equilibrium if \(c_0<0\) (i.e., \(R_0>1\));

- (2)There is a unique endemic equilibrium if$$\begin{aligned} (b_0<0\;\;\; \mathrm{and}\;\;\; c_0=0)\;\;\; \mathrm{or} \;\;\; (b_0<0,\;\;\; c_0>0\;\;\;\mathrm{and}\;\;\; b^{2}_0-4a_0c_0=0) ; \end{aligned}$$(14)
- (3)There are two endemic equilibria if$$\begin{aligned} b_0<0,\;\;\; c_0>0\;\;\;\mathrm{and}\;\;\; b^{2}_0-4a_0c_0>0; \end{aligned}$$(15)
- (4)
There are no endemic equilibria otherwise.

### **Theorem 2**

If \(R_0<1\), there exists an endemic equilibrium if conditions (14) are satisfied; two endemic equilibria if conditions (15) are satisfied; and no endemic equilibria otherwise. If \(R_0>1\), then there exists a unique endemic equilibrium.

### *Remark 1*

## 5 Bifurcation analysis

In this section we show that the sub-threshold occurrence of multiple endemic equilibria, as stated in Theorem 2, is the result of a transcritical backward (subcritical) bifurcation at \(R_0=1\). This will also provide insight on the local stability properties of the endemic equilibria emerging from the bifurcation.

**v**and

**w**denote the left and right eigenvectors, respectively, corresponding to the null eigenvalue of the Jacobian matrix of system (1) evaluated at \({\mathbf x}_0\) for \(\xi =0\).

### **Theorem 3**

If \(A_0<0\), then the malaria model (1) exhibits a backward bifurcation at \(R_0=1\). If \(A_0 > 0\), then the bifurcation is forward.

### *Remark 2*

## 6 Numerical simulations

*p*, increases \(R_0\). Figure 3 shows numerical simulations of the malaria model (1) for different values of

*p*. The curve for \(p=1\) shows the result for the assumption used in most malaria models: that malaria infection makes no difference to the attractiveness of humans. We see here that increasing

*p*decreases the equilibrium prevalence of infectious humans, contrary to the effect of

*p*on \(R_0\), but does not affect the equilibrium prevalence of infectious mosquitoes.

Figure 7 shows that the effects of simultaneously increasing *p* and \(\varphi _E\) on \(R_0\) are consistent with the one-dimensional sensitivity analyses shown in Figures 2 and 4. Additionally, as *p* increases, the dependence of \(R_0\) on \(\varphi _E\) increases.

## 7 Discussion and conclusions

We developed and analysed a simple malaria model to investigate the impact of relaxing three assumptions that mathematical models of malaria commonly make. We allowed the relative attractiveness of infectious humans (as compared to susceptible humans) to mosquitoes to increase; the feeding frequency of exposed mosquitoes (as compared to susceptible mosquitoes) to decrease; and the feeding frequency of infectious mosquitoes (as compared to susceptible mosquitoes) to increase.

We derived the basic reproductive number, \(R_0\), that provides a threshold condition for when the disease-free equilibrium loses stability. We showed that in common with many similar models of malaria, there is a transcritical bifurcation at \(R_0=1\), that can be forward or backward depending on the parameter values, and is always forward if there is no disease-induced death rate. We provided threshold conditions for the direction of the bifurcation and the existence of zero, one or two endemic equilibrium points.

As may be expected, we showed that as the relative attractiveness of infectious humans, *p*, increases, \(R_0\) increases. However, we also showed the surprising result that as *p* increases, the equilibrium proportion of infectious humans decreases, even as \(R_0\) increases. This is because mosquitoes repeatedly bite infectious humans so they are less likely to pass the infection to susceptible humans, and therefore the same proportion of infectious mosquitoes leads to a lower proportion of infectious humans for higher values of *p*.

We showed that as the relative biting rate of exposed mosquitoes, \(\varphi _E\), increases, \(R_0\) and the equilibrium proportion of infectious humans and mosquitoes decreases. This is reasonable because as the biting rate of exposed mosquitoes increases, their mortality increases so fewer exposed mosquitoes are likely to survive to become infectious.

We finally showed that varying the relative biting rate of infectious mosquitoes, \(\varphi _I\), has no impact on \(R_0\) because the increased biting rate is cancelled by the shorter life span of the infectious mosquitoes. Correspondingly, varying \(\varphi _I\) has no impact on the equilibrium proportion of infectious humans. However, increasing \(\varphi _I\) leads to a substantial decrease in the equilibrium proportion of infectious mosquitoes because of their higher death rate. Importantly for malaria transmission, these fewer infectious mosquitoes are able to maintain transmission to humans at a higher level than if their biting rate was lower. These analyses of a simple malaria model show that common assumptions around the relative attractiveness of infectious humans and the relative biting rates of exposed and infectious mosquitoes can have substantial and counter-intuitive effects on malaria transmission dynamics.

## Notes

### Acknowledgments

H. A. thanks the financial support of the University Institute of Technology of Ngaoundere through the research mission in the year 2016. The work of B. B. has been partially supported by local grant 2015-2016 ‘Analysis of Complex Biological Systems’ and has been performed under the auspices of the Italian National Group for the Mathematical Physics (GNFM) of National Institute for Advanced Mathematics (INdAM). N. C. has been supported by the Bill and Melinda Gates Foundation through grant OPP1032350.

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