An epidemiological model for mosquito host selection and temperature-dependent transmission of West Nile virus

Among the 325 municipalities in Greece during the period 2010–2021, WNV events, defined as the occurrence of at least one laboratory-confirmed human WNV case during a specific year, were reported in 154 (47%) municipalities, while the remaining 171 did not report any WNV case. WNV events were reported for a period ranging from one to eight years: 54 (35%) municipalities reported laboratory-confirmed WNV cases in only one year, 38 (25%) in two years, 30 (19%) in three years, 12 (8%) in four years, 10 (6%) in five years, 6 (4%) in six years, 1 (1%) in seven years, and 3 (2%) in eight years. This means that in 60% of the positive areas (82 municipalities out of 154), WNV appeared at most for two years, in 27% (42 out of 154) between three and four years, and in the remaining 13% (20 out of 154) for five years or more. Considering the total number of reported laboratory-confirmed human WNV cases across the twelve years (Fig. 1), in approximately 50% of the positive municipalities (78 out of 154), at most 4 cases were reported: 1, 2, 3, and 4 WNV cases were reported in 24, 32, 11, and 11 municipalities, respectively. Overall, 39 municipalities recorded a number of WNV cases ranging from 5 to 10 (third quartile), 34 a number ranging from 11 to 46, while the remaining 3 municipalities recorded a number of WNV cases equal to 56, 71 and 94.

Model evaluation and comparison with MIMESIS

We investigated the ability of the MIMESIS-2 model to correctly identify the occurrence of WNV events, both in space and time, and its capacity to quantify the annual number of human WNV cases and the timing of the first WNV event in the year. The performance of many quantities of interest, such as the severity and timing of occurrence of human WNV cases, was also compared output from the original MIMESIS model²⁶.

Occurrence of WNV events

Starting with the spatial analysis, we considered the fit of the model to replicate the observed 385 WNV events out of 3,900 (325*12) possible events across municipalities. MIMESIS-2 was able to correctly identify 356 of them, generated only one false alarm, and correctly modelled 3,514 true negatives.

The performance of MIMESIS-2 was then evaluated according to four indices: the probability of detection (POD), false alarm rate (FAR), miss rate (MIS), and critical success index (CSI), described in the Methods section. For the POD, MIS and CSI, we considered the 154 municipalities with at least one reported and laboratory-confirmed human WNV case over the 12-year period, while for the FAR, we considered the 153 municipalities where at least one human WNV case was modelled over the same period. We split the (0.0–1.0) index interval into five equally sized bins to derive for each index, the fraction of municipalities falling into each bin. Both the POD and CSI were above 0.8 for 139 municipalities out of 154, while the MIS was below 0.2 for 142 municipalities (out of 154) and the FAR was always below 0.2, with one false alarm produced in a municipality where WNV events were observed in eight out of twelve years (Table 1).

We also analysed how the model performed in different years by studying the multiannual evolution of the indices. Both the aggregated POD and CSI were equal to 0.92, with annual variations ranging from 0.72 (2021) to 1 (2011 and 2014). The aggregated MIS was 0.08, ranging from 0.0 (2011 and 2014) to 0.28 (2021). The FAR was virtually 0, being always equal to 0.0, with the only exception being 2017, when it was 0.1 (Table 2).

Magnitude and timing of WNV events: performance and comparison with MIMESIS

To evaluate the ability of MIMESIS-2 to capture the magnitude and timing of WNV events, we first considered the discrepancy between the overall number of observed and modelled WNV cases during the 12-year period for each municipality. Out of the 153 municipalities where at least one case was modelled across the 12 years, 76 (50%) had at most 4 modelled cases of WNV: 1, 2, 3, and 4 WNV cases were modelled in 22, 31, 13, and 10 municipalities, respectively. In 42 municipalities, the number of modelled cases ranged from 5 to 10 (which, as for the observed WNV cases, coincided with the third quartile), and in 32 municipalities, the number ranged from 11 to 47, while the remaining 3 municipalities had 55, 70, and 99 modelled cases (Fig. 1).

The MIMESIS-2 model closely replicated the total number of laboratory-confirmed WNV cases during the 12-year period. When considering only the 154 municipalities that recorded at least one WNV event during the considered period (excluding the true negatives), for 140 of them, the modelled number of cases fell within a ± 10% error range of the observed value, whereas for 149 the modelled number of cases fell within the ± 25% error margin. Only two municipalities showed a percent error above 50%. These were particular instances where only one WNV case was reported throughout the considered period, while MIMESIS-2 fitted zero human cases. For the original MIMESIS model, 63 and 84 municipalities fell within the ± 10% and ± 25% error margins, respectively, while 31 municipalities—mainly those where few cases were observed— had a relative error ≥ 100% (Fig. 2).

To further evaluate the bias of the model across all municipalities and years, we explored the difference between the yearly modelled and observed human WNV cases both with MIMESIS-2 and the original MIMESIS (IH_MOD-IH_OBS) across municipalities. In MIMESIS-2, we excluded 3,514 true negative cases to avoid distorted conclusions. For the remaining 386 cases, the mean bias was -0.04 indicating a possibly unbiased model, with the standard deviation (SD) of the residuals equal to 0.66 (original MIMESIS: mean bias 0.33, SD 2.07, after removing 3,387 true negatives) (Supplementary Fig. 1).

Across the 325 municipalities and the 12 years, 385 WNV events were observed, while on 3,515 occasions, no laboratory-confirmed human WNV cases were reported; on 162 occurrences, 1 case was reported, and on 67 and 39 occasions, 2 and 3 cases were reported, respectively. The maximum yearly number of human WNV cases observed in a single municipality was 38. Considering the modelled human WNV cases with MIMESIS-2, the distribution of the 356 hits ranged between 1 and 37 modelled cases, closely mimicking the distribution of the observed cases, since 1, 2 and 3 human WNV cases were modelled on 129, 72 and 37 occasions, respectively. For the 29 misses, the observed numbers of human cases were 1 (24 times), 2 (3 times), or 3 (2 times). The only false alarm was produced in the Pellas municipality, where WNV events were observed in 8 out of the 12 years.

We evaluated the timing of the first occurrence of WNV in humans for any municipality and year. Ignoring the municipalities with zero cases, the observed and MIMESIS-2-modelled first WNV cases occurred between weeks 22 and 44 and weeks 24 and 36, respectively. Modelled values tended to be dispersed around the observed ones: excluding the 3514 true negatives, 290 (75.13%) of the remaining 386 cases fell into the ± 4-week error margins from the observed cases (Fig. 2). This translated into a much lower bias of the week of first appearance (WY_MOD-WY_OBS) with respect to MIMESIS (Supplementary Fig. 2).

Case study: The Pellas municipality

In addition to presenting the overall performance of the model throughout different years and Greek municipalities, we highlight here the capacity of the model to capture population-specific behaviour and epidemiological features, such as the force of infection, that is, the rate at which susceptible humans, birds, and mosquitoes become infected, by presenting a single municipality case study for the municipality of Pellas. The Pellas municipality had the highest number of observed WNV cases over the 12-year period with a total of 94 human WNV cases, 38 in 2010, 16 in 2018, and 13 in 2021, no cases from 2014 to 2017, and between 4 to 8 cases in the remaining years.

We considered the impact arising from the changes in parameters defining the forces of infection. In addition to the introduction of bird (({psi }_{B})) and human (({psi }_{H})) host selections, changes included modifications for the mosquito-to-bird (({p}_{M})) and bird-to-mosquito (({p}_{B})) probabilities of transmission, whose values were made temperature-dependent following Vogels et al.²¹, and the replacement of the mosquito-to-bird (({varphi }_{B})) and mosquito-to-human (({varphi }_{H})) ratios with their dynamic counterparts, ({N}_{M}/{N}_{B}) and ({N}_{M}/{N}_{H}), respectively (Fig. 3). We used the May–October period for the 12 years that were considered, because this is the part of the year when Culex pipiens mosquitoes are reproductively active and the majority of human WNV cases are reported. In each year of the 12-year period, ({p}_{M}) started from 0.02, reached its peak—ranging from 0.16 to 0.25—in midsummer, and then decreased to the initial values (in the original model, ({p}_{M}=0.9)). Similarly, ({p}_{B}) started from 0.28, peaked in the same time interval—with maximal values ranging from 0.51 to 0.56—and then returned to the initial values (in the original model, ({p}_{B}=0.125)). Additionally, the dynamic specifications of ({varphi }_{B}) and ({varphi }_{H}) were shown to play an important role. Whereas in MIMESIS ({varphi }_{B}=30), in MIMESIS-2 the values started at approximately 8.6 and peaked in late summer when more human WNV cases are reported, reaching values of approximately 57, before decreasing to values ranging from 31.15 to 41.60 in late October. In MIMESIS, ({varphi }_{H}) was calibrated at the municipality level, and for Pellas municipality, it was 0.0001, whereas the dynamic counterpart in MIMESIS-2 showed a temporal evolution with a shape (but different scale) similar to that of ({varphi }_{B}), starting from values of approximately 1, peaking in late summer to values of approximately 7, and then decreasing to values of approximately 4 in late October.

Changes in these parameters enter into the expression for the forces of infection. It is of major practical interest to investigate how the values for the forces of infection resulting from MIMESIS-2 may vary for different values of the relative abundance of the vectors with respect to the corresponding carrying capacity and the temperature in different months (Fig. 4). As expected, all forces of infection increased with both the temperature and the relative abundance of the infectious vertebrate hosts. It is worth noting the importance of day length, as this affects the fraction of nondiapausing mosquitoes, ({delta }_{M}), and causes the forces of infection, all other things being equal, to be potentially higher in June and July than in the other months. However, in these two months, the modelled forces of infection tend to be smaller than those in August due to the lower abundance of infectious hosts.

The bird-to-mosquito force of infection, ({uplambda }_{BM}), took values on the order of 10^–4, with possible peaks of approximately 7 × 10^–4 in the case of high temperature and high prevalence of birds in June and July, which were nevertheless not reached due to a low abundance of infected birds in that period. Considering the months of July and August 2021 for illustrative purposes, the resulting modelled values were 1.20 × 10^–4 and 2.19 × 10^–4, respectively, with the increase in August explained by a higher abundance of infected birds in that period. It is worth noting that if the infection across birds had a lead period of two weeks, the resulting ({uplambda }_{BM}) in July would become 3.91 × 10^–4 (+ 226%), while an increase in the average temperature in August by 1 °C would result in ({uplambda }_{BM})= 2.36 × 10^–4 (+ 8%). The mosquito-to-bird, ({uplambda }_{MB}), and mosquito-to-human, ({uplambda }_{MH}), forces of infection showed similar qualitative behaviours, albeit at different scales, and in this case, they were higher in August due to a higher prevalence of infected Culex mosquitoes in that month. More specifically, ({uplambda }_{MB}) equalled 1.06 × 10^–3 and 1.22 × 10^–3 at the end of July and August, respectively, and an expected two weeks for the infection of mosquitoes would result in ({uplambda }_{MB})=4.15 × 10^–3 (+ 292%) at the end of July, while an increase in the average temperature in August by 1 °C would result in ({uplambda }_{MB})= 1.31 × 10^–3 (+ 7%) at the end of August. Finally, ({uplambda }_{MH})=2.86 × 10^–6 at the end of July, while ({uplambda }_{MH})=3.26 × 10^–6 at the end of August, with the anticipation of the infection among mosquitoes by two weeks resulting in ({uplambda }_{MH})=1.12 × 10^–5 (+ 290%) and an increase in the average August temperature by 1 °C leading to ({uplambda }_{MH})=3.51 × 10^–6 (+ 8%). It is worth recalling that since we calibrated the model on the number of reported laboratory-confirmed human WNV cases, ({uplambda }_{MH}) represents the rate at which susceptible humans contract infection and become symptomatic leading to a recorded human WNV case.

We explored changes in the populations of infectious hosts and the total population number for both mosquitoes and birds over 2010–2021 for the period spanning from May to October (Figs. 5 and 6). The population of infected mosquitoes (({I}_{M})) was initialised by calibration (see the Methods section). Each year, after a short period in which the population of infected mosquitoes slightly decreased due to a very small number of infectious birds (({I}_{B})) that prevented the infection from spreading, it started growing substantially during summer, reaching its peak in late summer, coinciding with the period when most human cases were recorded. The observed increase in ({I}_{M}) was combined with the growth ({I}_{B}) at approximately the same time (with a slightly anticipated peak), which had an amplification effect on the spread of the infection. Both ({I}_{M}) and ({I}_{B}) showed significant yearly variation, with higher modelled numbers in years where more human WNV cases were reported. The modelled total population of mosquitoes (({N}_{M})) did not show significant interannual variability, always peaking in late summer. Finally, the overall population of birds (({N}_{B})) did not show any variability in the first part of the year, when an increase due to immigration and offspring generation was observed, whereas it had a moderate interannual variability in the second half of the year. These differences may be due to heterogeneous numbers of observed infected, dead and immune birds.

Comparison of these population dynamics with those of MIMESIS revealed interesting patterns (Fig. 6). Considering the relative number of mosquitoes in MIMESIS-2 with respect to MIMESIS, the populations in MIMESIS tended to grow faster due to a higher mosquito carrying capacity (({K}_{M})) in the original model (({K}_{M}) ≈ 8.3 × 10⁵ in MIMESIS versus ({K}_{M}) ≈ 2.4 × 10⁵ in MIMESIS-2), resulting in a decrease in the ratio between the amounts modelled by MIMESIS-2 and the ones modelled by MIMESIS. Significant interannual variability could be seen in the first part of the year for infectious mosquitoes, where different initial calibration values played an important role. For the populations of birds, until midsummer, the overall number modelled by MIMESIS-2 tended to be approximately 1/4 that of MIMESIS, while as of July, different patterns were observed due to the higher mortality of birds in the original MIMESIS model. In years with higher virus spread, higher mortality was reflected in a sharper decrease in bird populations; therefore, the ratio between the population modelled by MIMESIS-2 and that modelled by MIMESIS increased up to approximately 0.6 (2010).

Source: Ecology - nature.com