Section: Scientific Foundations
Dealing with heterogeneity using Complex Model
Modeling and analysis of epidemiological models
where represents the concentration of susceptible individuals or target cells, represents the different class of latent, infectious and removed individuals. The matrix C is a nonzero k×n nonnegative matrix, is a positive vector, P denotes a linear projection, A is a stable Metzler matrix and denotes a scalar product in . The function describes the recruitment (or the demography) of susceptible individuals or cells and the quantity represents the infection transmission. For some diseases, a bilinear infection transmission function is not adequate so we have to replace in equation (3 ) the expression Cy by a more general non-linear incidence function Cf(y) . The parameter u takes only the value 0 or 1.
The model (3 ) represents either the transmission of a directly transmitted disease (i.e transmitted by adequate contact, Ebola, Tuberculosis, ...), in this case u = 0 , or represents the intra-host dynamics of a parasite with target cells. To illustrate this claim we will give two examples.
The system (3 ) can represent the so called DI, SP or DISP models. In the studies of the transmission dynamics of HIV, two fundamental hypotheses for variations in infectiousness have been made. In the staged-progression (SP) hypothesis, the infected individuals sequentially pass through a serie of stages, being highly infectious in the first few weeks after their own infection, then having low infectivity for many years, and finally becoming gradually more infectious as their immune system breaks down and they progress to AIDS. Based on other clinic findings and blood serum level studies, another hypothesis is the differential infectivity (DI) hypothesis, where infected individuals enter one of several groups j (j = 1...n ) with probability j , depending on their infectivity, and stay in that group until they develop AIDS. If we denote by S the density of susceptible individuals, Ii the density of the different classes of infectious individuals, the DI model can be represented by a compartmental model:
which gives the differential equation
where is an input flow (or a recruitment rate) which is supposed to be constant, is the natural death rate of the population. For each j the parameter j is the contact rate, i.e., the rate at which susceptibles meet infectious individuals belonging to the class j , the parameter j is the disease-related death rate of the class j and
Similarly the SP model can be represented by
The parameter j denotes the fractional rate of transfer of infected from the stage j to the stage j + 1 . The dynamical progression of the disease can be represented by the differential equation:
The DISP is the combination of these two structures. These models are easily put under the general form. (3 ).
This general form can also represents intra-host models : We sketch the example of malaria  . We give a brief review of the biological features of malaria. Malaria in a human begins with an inoculum of Plasmodium parasites (sporozoites) from a female Anopheles mosquito. The sporozoites enter the liver within minutes. After a period of asexual reproduction in the liver, the parasites (merozoites) are released in the bloodstream where the asexual erythrocyte cycle begins. The merozoites enter red blood cells (RBC), grow and reproduce over a period of approximately 48 hours after which the erythrocyte ruptures releasing daughter parasites that quickly invade a fresh erythrocyte to renew the cycle. This blood cycle can be repeated many times, in the course of which some of the merozoites instead develop in the sexual form of the parasites : gametocytes. Gametocytes are benign for the host and are waiting for the mosquitoes. An important characteristic of Plasmodium falciparum , the most virulent malaria parasite, is sequestration. At the half-way point of parasite development, the infected erythrocyte leaves the circulating peripheral blood and binds to the endothelium in the microvasculature of various organs where the cycle is completed. A measurement of Plasmodium falciparum parasitaemia taken from a blood smear therefore samples young parasites only. Physician treating malaria use the number of parasites in peripheral blood smears as a measure of infection, this does not give the total parasite burden of the patient. Moreover antimalarial drugs are known to act preferentially on different stages of parasite development. Hence to model the dynamics of parasitized erythrocytes, it is natural to introduce different classes. Then we propose the following model
where the variable x denotes the concentration of uninfected RBC, the variable yj is the concentration of parasitized red blood cell (PRBC) of class j , and m is the concentration of the free merozoites in the blood. The example of malaria gives an example where stages in modeling are created for biological reasons. We have seen before that continuous delays are important to be modeled. The process of converting time-delay integro-differential equations in a set of ODE is coined by MacDonald  as the linear chain trick. In other community this is also known as the method of stages. Actually any distribution can be approximated by a combination of stages in series and in parallel (Jacquez). This process consists to insert stages in the model. This is an example of stages created to take into account a behavior. This added stages have no biological meaning. Our general model is also well suited for this process.
The general model (3 ) can take into account the case of different strains for the parasites and can be adapted to cope with vector transmitted diseases. Then we have a building block to model complex systems. System (3 ) describes the basic model which can be extended, by introducing interconnections of blocks of the form (3 ), to describe more complex systems : more classes of susceptible can be introduced, the recruitment of susceptible individuals can be replaced by an output of an explicit model of the population dynamics, each sub-system describes what happens in a patch, inflows and outflows can be introduced to model the population movement between patches, different strains for the pathogen can be introduced, others systems can bring input in these models (e.g. the immune system) ...
This general form will be used to model some well-identified diseases for which we have data and expert collaborators (e.g. malaria, dengue, Ebola ...). This form has to be tailored to the particular case considered. For example the matrix A represents connections and the structure of this matrix A (triangular, Hessenberg, sparse ...) depends on the disease.