Mathematical epidemiology:Past,present,and future

We give a brief outline of some of the important aspects of the development of mathematical epidemiology.

General Compartmental Epidemic Models

The age of infection approach introduced by Kermack and Mckendrick in 1927 gives a unified way of describing and analyzing a variety of epidemic models, including models with multiple stages, treatment, and heterogeneous mixing. The author gives a description of the main results for such models, emphasizing the use of the final size relation to estimate the size of the epidemic.

A final size relation for epidemic models of vectortransmitted diseases

We formulate and analyze an age of infection model for epidemics of diseases transmitted by a vector,including the possibility of direct transmission as well.We show how to determine a basic reproduction number.While there is no explicit final size relation as for diseases transmitted directly,we are able to obtain estimates for the final size of the epidemic.

Some models for epidemics of vector-transmitted diseases

Vector-transmitted diseases such as dengue fever and chikungunya have been spreading rapidly in many parts of the world.The Zika virus has been known since 1947 and invaded South America in 2013.It can be transmitted not only by(mosquito)vectors but also directly through sexual contact.Zika has developed into a serious global health problem because,while most cases are asymptomatic or very light,babies born to Zika-infected mothers may develop microcephaly and other very serious birth defects.We formulate and analyze two epidemic models for vector-transmitted diseases,one appropriate for dengue and chikungunya fever outbreaks and one that includes direct transmission appropriate for Zika virus outbreaks.This is especially important because the Zika virus is the first example of a disease that can be spread both indirectly through a vector and directly(through sexual contact).In both cases,we obtain expressions for the basic reproduction number and show how to use the initial exponential growth rate to estimate the basic reproduction number.However,for the model that includes direct transmission some additional data would be needed to identify the fraction of cases transmitted directly.Data for the 2015 Zika virus outbreak in Barranquilla,Colombia has been used to fit parameters to the model developed here and to estimate the basic reproduction number.

A co-interaction model of HIV and syphilis infection among gay,bisexual and other men who have sex with men

We developed a mathematical model to study the co-interaction of HIV and syphilis infection among gay,bisexual and other men who have sex with men(gbMSM).We qualitatively analysed the model and established necessary conditions under which disease-free and endemic equilibria are asymptotically stable.We gave analytical expressions for the reproduction number,and showed that whenever the reproduction numbers of sub-models and co-interaction model are less than unity,the epidemics die out,while epidemics persist when they are greater than unity.We presented numerical simulations of the full model and showed qualitative changes of the dynamics of the full model to changes in the transmission rates.Our numerical simulations using a set of reasonable parameter values showed that:(a)both diseases die out or co-exist whenever their reproduction number is less than or exceed unity.(b)HIV infection impacts syphilis prevalence negatively and vice versa.(c)one possibility of lowering the co-infection of HIV and syphilis among gbMSM is to increase both testing and treatment rates for syphilis and HIV infection,and decrease the rate at which HIV infected individuals go off treatment.

Modelling and simulating Chikungunya spread with an unstructured triangular cellular automata

In this work we propose a mathematical model to simulate Chikungunya spread;the spread model is implemented in a C++ cellular automata code defined on unstructured triangular grids and space visualizations are performed with Python.In order to simulate the time space spread of the Chikungunya diseases we include assumptions such as:heterogeneous human and vector densities,population mobility,geographically localized points of infection using geographical information systems,changes in the probabilities of infection,extrinsic incubation and mosquito death rate due to environmental variables.Numerical experiments reproduce the qualitative behavior of diseases spread and provide an insight to develop strategies to prevent the diseases spread.

A singular perturbation approach to epidemics of vectortransmitted diseases

In vector-borne epidemic models there is often a substantial difference between the vector and host time scales.This makes it possible to use the quasi-steady-state to obtain final size relations.