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The Liverpool Malaria Model (LMM)

In order to assess the occurrence of malaria in West Africa an existing model from the University of Liverpool is applied. The so-called Liverpool Malaria Model (LMM) simulates the spread of malaria at a daily resolution using daily mean temperature and 10-day accumulated precipitation (Hoshen and Morse 2004).

Fig. 1: Illustration of various components of the Liverpool Malaria Model (LMM).

The gonotrophic cycle

The model first simulates the egg development within the female mosquitoes, the so called gonotrophic cycle. The ensuing gonotrophic cycle is temperature dependent and starts at a certain temperature threshold that is dependent on humidity conditions (Detinova 1962). Since humidity is often reliant on precipitation a precipitation threshold is defining dry and humid weather conditions. Using the "degree-day" concept (cp. Glossary) the model is calculating the daily progress of the egg development within the female mosquito. At the end of the gonotrophic cycle the mosquitoes oviposit at suitable breeding sites and are immediately starting a new gonotrophic cycle.

Egg deposition and immature mosquitoes

The egg deposition is dependent on open water bodies that are mostly filled by precipitation events. At present an accurate simulation of open water bodies as well as the number of laid eggs is infeasible. These problems were leading to the simplification of the process of the egg deposition. The oviposition rate is roughly assumed to be proportionate to both the number of ovipositing mosquitoes and to the decadal rainfall (RRΣ10d). The model approach implies that no permanent water sources like rivers or lakes are in the vicinity. Also agricultural activities are neglected, e.g. rice irrigation is affecting the availability of fresh water and hence the spread of malaria (e.g. De Plaen et al. 2004). Following the egg stage the mosquito larvae are developing. After a certain time period the larvae pupate and finally the maturation takes place. The immature mosquito stage is simplified in the LMM. The process is simply finished after 15 days, and the model assumes a rainfall dependent per diem survival rate for the immature mosquitoes (s). The probability of survival varies between about 50% for dry conditions and nearly 100% for periods with abundant rainfall: s=(RRΣ10d+1)/(RRΣ10d+2).

The sporogonic cycle

The malaria transmission starts in the model when mosquitoes are biting malaria infectious humans. The course of the infection in the mosquito vector is again modelled via the degree-day concept. It has long been recognised that the development of sporozoites is temperature dependent (e.g. Detinova 1962; Snow et al. 1997). The minimum temperature for the development of the malaria parasite varies experimentally and also among mosquito species (Epstein et al. 1998). There is an uncertainty about the value of the temperature threshold. Lindsay and Birley (1996) therefore conclude, that the parasite development ceases below temperatures between 14.5 and 16°C for Plasmodium vivax and between 16 and 19°C for Plasmodium falciparum. The setting of the threshold is important when the malaria is modelled in areas with temperatures in the range of the minimum temperature (e.g. in the highlands of East Africa). For temperatures well above the threshold the length of the sporogonic cycle is much less dependent upon the setting of the minimum temperature (cp. Fig. 2). Regarding the sporogonic cycle the LMM was originally set by a temperature threshold of 18°C (Hoshen and Morse 2004). However, modelled temperatures or data from weather station are unlikely to record conditions in the microhabitats where vectors spend most of their time (Hay et al. 1996; Kovats et al. 2001). By resting more climatically stable and warmer houses Anophelines vectors may avoid cold temperatures and thus the restrictions concerning the progress of the parasite development (e.g. Epstein et al. 1998; Reiter 2001; Koenradt et al. 2006). Therefore the effect of altitude might be partly compensated, when mosquitoes stay in heated houses (Malakooti et al. 1998). For these reasons the use of 16°C as a temperature threshold for parasite development is decided. Based on this setting the duration of the sporogonic cycle lasts about 57 and 13 days at temperatures of 18 and 25°C, respectively (cp. Fig. 2).

Fig. 2: The effect of temperature on the duration of the sporogonic cycle in days with regard to different sporogonic temperature thresholds.

Mosquito survival

Due to the longevity of the parasite development the mosquito survival probability is of great importance. The age structure of Anopheline females and survival rate exerts a strong influence on the reproduction rate of the mosquito population and the spread of the malaria parasite. Hence the vector survivorship is of paramount ecological importance for the distribution of malaria (e.g. Service 1976; Clements und Paterson 1981; Lee et al. 2001; McKenzie et al. 2002; Scholte et al. 2003). The survival depends on characteristics of the mosquito species, the activities of individuals, climate, the incidence of parasites, and predators (Boyd 1949) and the age of the mosquito (e.g. Samarawickrema 1967). Most of these factors are elusive and are only indirectly taken into account in malaria models. The LMM is only considering the weather impact on vector survivorship. However, the relationship between mosquito survival and humidity is unclear (Lindsay and Birley 1996), and therefore the humidity is not taken into account in the numerical LMM setting.
Various mosquito survival probability schemes have been developed with regard to the modelling of malaria. In the LMM five different schemes are implemented, these are: the so called Lindsay-Birley, the Martens I, the Martens II, the Bayoh, and the Bayoh-12.5% scheme (Fig. 3). Initially, the LMM was set by the Lindsay-Birley scheme (Hoshen and Morse 2004 and 2005), that assumes a constant mosquito survival of 50% per gonotrophic cycle. However, the application of this scheme is not possible at very high temperatures.
Due to the scarce information regarding the temperature dependence of the mosquito survival the literature (e.g. Craig et al. 1999; Hay et al. 2000a) refers to studies published by Martens (Martens et al. 1995a and 1995b; Martens 1997). Martens (1997) states: "Relying on data reported by Boyd (1949), Horsfall (1949), and Clements & Patterson (1981), we assume a daily survival probability of 0.82, 0.90 and 0.04 at temperatures of 9°, 20° and 40°C, respectively, expressed as: p=exp(-1/(-4.4+1.31 T-0.03 T2.
The Martens I scheme was obviously generated as a polygon connecting the quoted three data points in the T-pd-diagram (Fig. 3). The formula provided by Martens (1997) was not considered and is leading to different survivorships, the so called Martens II scheme. The main difference between the Martens I and Martens II scheme is the more smoothly decrease of the survival probabilities for higher temperatures.
Bayoh (2004) observed the survival and mortality rates of Anopheles gambiae sensu stricto in environmental chambers at combinations of temperatures from 0 to 45°C at 5°C intervals and relative humidity's of 40%, 60%, 80%, and 100%. Using the data of these experiments and assuming an exponential model of mortality it is possible to derive daily vector survival probabilities. The values are used to define a polygon regarding Anopheles gambiae sensu stricto survivorship in the laboratory, the so called Bayoh scheme (Fig. 3). However, vector survival is higher in captivity than in the wild (e.g. Clements and Paterson 1981) and hence has to be reduced. The shift-off is arbitrarily set to 12.5%. The probabilities of the Bayoh-12.5% scheme compare well with field observations. The Bayoh-12.5% scheme agrees also fairly well with the mean daily survival of 84.6% found by Kiszewski et al. (2004).
As shown here, all the presented schemes regarding mosquito survival are not fully satisfying. Therefore the incorporation of upcoming new information is essential in future refinements of the LMM.

Fig. 3: Illustration of different schemes regarding the daily mosquito survival (in %) against the daily mean temperature (in °C). Displayed is also the data basis of the two Martens schemes (the three black dots).

Malaria infection of humans

After the completion of the sporogonic cycle the mosquitoes are able to transmit the parasite to a human. When an infected female mosquito is biting one of the 100 humans of the model the parasite is transmitted with a certain probability. The so called inoculation rate is set in the LMM by the default value of 50 %. All the infected human hosts are entering the latent phase. The parasite development within the human host lasts about two weeks (Hoshen and Morse 2004). In the model infected humans are infectious at day 15 and are at that time able to infect mosquitoes with the male and female gametocytes until these sexual forms of the parasite vanish. Humans becoming infectious increase the so-called malaria prevalence, i.e. the proportion of the 100 humans that are carrying gametocytes. Malaria clearance is a slow process sometimes continuing several months or even years (Sama et al. 2004). In the model the daily infection survival rate (d) is 0.9716 and is consistent with malariotherapy (Hoshen and Morse 2005). The exponential decay reflects a probability of about 90% of an individual clearing an untreated infection within 80 days.
In areas of high infection pressure the prevalence is often constant for a certain time. The equilibrium condition is reached, when all the modelled humans are either infected or infectious. During that time the numerous infectious mosquitoes are again infecting every human clearing the parasite prevalence. Due to the fixed recovery rate only 64.9% pass the latent period and will become infectious. That is the reason that for saturated conditions the malaria prevalence is not getting higher than 64.9%.

Summing up, the LMM is simulating the malaria transmission within the human and mosquito population and is only forced by two meteorological variables, daily mean temperature as well as daily precipitation amounts.

Sensitivity of the LMM

Various sensitivity experiments reveal that LMM is fairly sensitive to the setting of some of the model parameters. The proportion of the population that is carrier of the malaria parasite strongly depends on the applied mosquito survival scheme. In addition the malaria transmission as simulated by LMM strongly depends on the recover rate (r=0.0284) of humans. Furthermore, in areas where temperature is not a factor the simulated malaria transmission from mosquitoes to humans is mainly governed by the model rainfall multiplier that couples the precipitation with the oviposition of female mosquitoes and ultimately determines the size of the mosquito population. At high altitudes, the sporogonic temperature threshold, i.e. the minimum temperature for malaria parasite development in the mosquito, is of importance. Unlike in the model version described by Hoshen and Morse (2004) the LMM was parameterised by a different mosquito survival scheme and a sporogonic temperature threshold of 16°C.

  • Original and interim LMM settings