(Lecture by Prof Eduardo Massad, Mathematical Models of Dengue Fever) This lecture touches on the topic of modeling for dengue fever, yellow fever and chikungunya. Geographical applications abound both for areas in Brazil and in Singapore. Vaccination strategies could depend on such models to acquire a more effective result, as this model suggests a parameter Pc, the critical proportion of a population to be vaccinated to prevent an epidemical outbreak of these respective viral diseases.
Limitations to the model are discussed, such as the temperature dependence of the parameters of the model are considered, and especially with regard to increasing global temperatures due to global warming. Description of models discussed in lecture Prof Massad first mentions the parameter Ro, the basic reproduction number, defined as the number of secondary infections produced by a single infective in an entire susceptible population (Macdonald, 1952). Ro= ma2bcexp-? nr ? wherem:host population, a:average daily biting rate of the vector, c:the mosquito’s susceptibility, µ:the vector mortality rate,n:the parasite extrinsic incubation period, in days,| r:the parasitemia recovery rate.
(Massad et. al. , 2001) However, this equation for Ro is valid for simple cases where there is only one vector or breed of mosquitoes, and one host (i. e. humans). The resulting equation derived for more complex systems is thus Ro= aNmaNHraexp-? ? b c? whereNm :number of female mosquitoes. a: daily biting rate female mosquitoes inflict on the human population. NH:number of humans. r:rate of recovery from parasitemia in the human cases c: probability that a mosquito gets the infection after biting an infective human.
B:probability that a human gets the infection after being bitten by an infective mosquito e-µ? :fraction of infected mosquito population that survives through the extrinsic incubation period, ? , of the parasite. Subsequent derivations of Ro based on dengue and yellow fever are obtained. Expressing the Ro for yellow fever as a function of Ro for dengue fever, we obtain the following equation. Royf= Rodengue? yf? denguebyfcyfbdenguecdengueexp-? (? yf-? dengue) Since there exists a vaccine against yellow fever, the objective to obtain a figure for proportion of the population, pc to be vaccinated can be obtained.
pc=1-1Royf A similar model to the dengue and yellow fever models can be obtained for chikungunya. Rochik= Rodengue? chik? denguebchikcchikbdenguecdengueexp-? (? chik-? dengue) Applications of dengue models to Singapore The temperature dependence of the parameters of the model could affect the accuracy of this model. Increasing global temperatures due to global warming affects countries such as Singapore. Figure 1: Increasing temperature profile and overall rise in dengue cases for Singapore between 1989 and 2005. Limitations.
There is a possibility that the parameters regarding mosquito characteristics may vary geographically as well, for example, the biting rates, infection rates by the vector of mosquitoes in Brazil versus the mosquitoes found in Singapore (Rocklov & A. , 2011). A particular case to note is the sudden drop in the number of cases of dengue fever in Year 2006 in Singapore, and Prof Massad speculates that the reason for the drop could due to the haze from Indonesian forest fires causing an increase in the mortality rate of the mosquitoes.
Conclusion Modeling of dengue fever, yellow fever and chinkungunya have been discussed, and these models give quantitative data to help decision makers analyse what kind of control strategies to implement to regulate the mosquito populations in a given area, and work has been done extensively by Prof Massad and his team in parts in Brazil such as Rio De Janeiro and Sao Paulo, as well as places in Asia such as Singapore.
Control strategies such as fogging and search and destroy methods to induce adulticide and larvicide of mosquitoes have been discussed in the lecture and the models give appropriate data to help decision makers adopt an appropriate strategy that has been effective in Singapore.
References Macdonald, G. (1952). The analysis of equilibrium in malaria. Tropical Diseases Bulletin, 49 , 813,-828. Massad, E. , Bezerra, C. F. , Lopez, L. F. , & da Silva, D. R. (2011).
Modeling the impact of global warming on vector-borne infections. Physics of Life Reviews 8 , 169-199. Massad, E. , Bezerra, C. F. , Nascimento, B. M. , & Fernandes, L. L. (2001). The risk of yellow fever in a dengue-infested area. Transactions of The Royal Society of Tropical Medicine and Hygiene, 95 , 370 -374. Rocklov, J. , & A. , W. -S. (2011). Climate change and vector-borne infections Comment on “Modeling the impact of global warming on vector-borne infections”. Physics of Life Reviews 8 , 204-205.