Extension to continuous time, EIR
Here, the ABM of mosquito host-seeking behaviour is linked to continuous-time compartmental modelling. This connects in situ mosquito behaviour to commonly measured quantifiers of malaria transmission, such as Entomological Inoculation Rate (EIR) and malaria incidence. A benchmark test for classical malaria models was conducted in [
10], where five dynamic models of malaria transmission were tested on the basis of established performances. The most basic Ross malaria model was found to be capable of reproducing the EIR experimental data satisfactorily. Indeed, more complex models tend to suffer from poor identification of parameters and may produce results inferior to simple but more robust modelling. Following [
10], the simple Ross model is considered, but utilized such that the complex factors (such as the LLIN coverage, household size or alterations of behaviour) are expressed via the ODE model parameters. That is, the regression functions from the previous section for the contact and mortality rates are substituted in place of the respective parameters in the Ross model:
$$\begin{aligned} di_h&= m\bar{a}b i_m(1 - i_h) - i_h r\nonumber \\ di_m&= \tilde{a} c i_h(1 - i_m) - \mu i_m , \end{aligned}$$
(17)
where
\(i_h\) and
\(i_m\) denote the fractions of infected humans and mosquitoes, correspondingly,
m stands for mosquito-to-human ratio,
b and
c are the probabilities of transmission during mosquito contact with the host,
\(\mu\) denotes the mosquito mortality rate, and
r stands for recovery rate for the humans. The difference to the conventional Ross model is also that the contact and death rates
\(\bar{a},\tilde{a}\) and
\(\mu\) are given by the response surfaces, fitted to various in-situ conditions. Indeed, two different contact rates, for infected
\(\overline{a}\) and uninfected
\(\tilde{a}\) mosquitoes are used in the case when alterations in mosquito behaviour is assumed. For the rest of the parameters,
m,
b,
c,
r, three sets of values were borrowed from [
10], corresponding to low, medium and high transmission settings, see Table
4. The integration of the Ross model is done for household size comprising 2, 4, 6, 8 and 10 individuals while applying 20, 40, 60, 80 and 100% LLIN coverage for each household size considered.
Table 3
Summary of the basic agent-based model parameters, [
20]
\(p_{net}\) | Probability of being blocked by the physical barrier created by the net |
\(p_{hut}\) | Probability of exiting the hut |
\(d_{50}\) | Range of repellent coverage |
\(\mu _p\) | Insecticide-induced death rate |
r | Intensity of repulsion |
\(t_{max}\) | Maximum host-seeking time (when confronted with the LLIN) |
\(\mu _e^G\) | Rate of increase of excito-repellency for An. gambiae |
\(\mu _e^A\) | Rate of increase of excito-repellency for An. arabiensis |
\(\alpha _G\) | Detoxification rate for An. gambiae |
\(\alpha _A\) | Detoxification rate for An. arabiensis |
\(t_{max}^{host}\) | Maximal time (in minutes) spent |
| Attempting to feed on protected host |
The quantities of interest are the equilibrium fractions of infected mosquitoes and humans. Note that the units for mortality and contact rates are the same in both the ABM and Ross model, given as a fraction of mosquito population subject to mortality (feeding) per day. The contact rate is understood here as the average number of bites taken by the mosquito diurnally.
Table 4
Sensitivity design table
Minimum | 2.7058 | 1.0718 | 0.8756 | 40/3 |
Maximum | 18.2942 | 14.9282 | 25.1244 | 80/3 |
The mosquito-to-human ratio
m is taken as a ratio of the number of humans to mosquitoes
\(N_m/N_h\), as given in [
10]. Each value of
m is combined with the three sets of the other parameters in Table
6, so nine pairs of equilibrium values of fractions of infectious humans
\(i_h^*\) and infectious mosquitoes
\(i_m^*\) were calculated. For each case, the response surfaces with respect to household size and LLIN protection can be now calculated.
Table 5
Sensitivity design table for the behavioral alteration
Minimum | 1.8934 | 5.8579 |
Maximum | 23.1066 | 34.1421 |
Table 6
Summary of parameter selections and mosquito densities
m from [
10] used for integration of the Ross model
Quantity | High transmission | Medium transmission | Low transmission |
m | 7.6 | 5.5 | 4.0 |
The most direct approach for estimating the overall malaria transmission in a population is by computing the Entomological Inoculation Rate (EIR) [
15]. EIR is commonly measured to quantify the intensity of an infected mosquito pool and its propensity to transmit malaria infection to human populace within a given time period. Conventionally, the EIR is measured per period of time: per night, monthly, seasonally or annually. The transmission patterns represented by the pair of EIR and Parasite Rate (PR) depend on a number of ecological, climatic and socioeconomic factors [
49]. Here, the simulation results are compared to the experimental results reported in [
50], where a trend curve together with the 95% confidence interval was created using data from 31 sites in Africa. At the time when the survey was published, the annual EIR varied from less than 1 to more than 1000 infective bites per person per year. The transmission patterns represented by the pair of EIR and Parasite Rate (PR) depend on a number of complex factors, such as ecology, climate and socio-economic development [
49]. By integration of the modified Ross model, the impact of partial population coverage with LLIN, alterations in mosquito behaviour and household size, on EIR and PR, can be quantified. These two factors are computed from equilibrium fractions of infectious mosquitoes
\(i_m^*\) and humans
\(i_h^*\), i.e., by the steady state of the Ross model. As EIR is defined as the product
\(mai_m^*\), a direct computation gives:
$$\begin{aligned} EIR= m \bar{a} i_m^* = \frac{\bar{a}\tilde{a} bcm - \mu r}{\mu b + \tilde{a} b c}, \end{aligned}$$
where
$$\begin{aligned} i_m^* = \frac{\bar{a}\tilde{a} bcm - \mu r}{\mu m \bar{a}b + \bar{a} \tilde{a} b c m}. \end{aligned}$$
As a result, three pairs of equilibrium EIR and malaria incidence
\(i_h^*\) correspond to each of the original selections of parameters given in [
10], see Fig.
8. In addition, the EIR and PR values for those LLIN and household values for which the regression models of
\(\bar{a},\tilde{a}, \mu\) were calibrated, can now be computed. These values, as continuous functions of LLIN, are added in Fig.
8. Figure
9 gives an example of the response surface of the EIR values as a function of household size and LLIN coverage. Respective figures for all the chemicals are given in Additional file
1.
Note that all the results presented in this paper are based on data from [
19]. Additional experimental data would improve the reliability of the results, especially for the behaviour of mosquitoes between the households. Given that similar data are available elsewhere, the approach allows general trends and response surfaces to be produced based on such data in an analogous way.