In spite of the effort to fight Malaria, the size of the phenomenon is not decreasing. Furthermore, the resistance to the widest used drugs has spread since 1960. Those drugs have the characteristic to be long lasting in the organism. Since using one drug is not enough, new strategies use a combination of two drugs to take advantage of the multiplicative effect of mutation probabilities. Since the probability of mutation to be resistant to one drug is nevertheless low, the probability to be resistant to two drugs can be thought to be the product of both probabilities.
The combination uses usually a drug with long half-life, protected with a short half-lived drug. When taking both the drugs, the decrease of the population is considerable. When the short-lived drug has disappeared, the long lasting one still remains. The remaining population of parasites in presence of the residual drug is statistically to small to show a mutation, and is eliminated by the long lasting drug.

I tried to simulate the evolution of the transmission of resistance to the two drugs entering in the combination, using a model of differential equations. First, I simulated the comportment of the population of parasites in presence of immune cells, and without drug, so I could estimate the time for the symptoms (3% of parasitemia) of the disease to emerge. I added then one drug at this moment (9th day). Has we could imagine, all the sensitive parasites disappeared, leaving only those resistant. Then the immune cells eliminated all the remaining parasites. The cumulative production of resistant gametocytes (those transmitted by the mosquito) was very high (concentration reached 1e6 /ml after 20 days).
Then I made a simulation with variable concentrations of two different half-lived and effectiveness drugs, and without immune cells. For a definite combination of concentrations, the result showed a total elimination of the parasites, without high production of gametocytes.

I tested the length of protection of the patient with drug combination, against new infestation.
With the effective concentrations of drugs found before, I simulated the production of resistant gametocytes according to the influence of a second injection of parasites by a mosquito, and the variation of the mortality of the immune cells.
For an injection of sensitive parasites, the result showed a protection against parasite’s new installation, and therefore, no production of resistant parasite, of about 5 days after the ingestion of medicaments, and a longer period for a very low immune cells’ mortality (0.1/day). If the injection takes place after the protection, during a low drug concentration period, it gives a higher production of resistant parasites than an early injection, during high concentration of drug.
With an injection of sensitive plus both simple-resistant parasites (2%), the protection is only about one day, depending mainly on the short half-lived drug. Furthermore, if the parasites’ injection takes place after this protecting period, the production is mainly composed of long lasting medicaments resistant, and double resistant parasites.
Often, the rising of the secondly injected parasites is only delayed by the sub lethal remaining drugs. So the immune cells population has time to decrease to its ground equilibrium concentration, taking more time to rise again to eliminate parasites when parasitemia becomes high for the second time. We notice that the creation of resistant parasites to the short half-lived drug remains rare, whereas the resistance for the long half-lived drugs takes often place, in the condition of experiment. This shows that this (simplistic) treatment is, theoretically, usable to cure patients in healthy regions, but must be improved when used in regions with re-infestation, even when resistance is not yet established, since the protection of a (long half-lived drug) by an other (short half-lived) is not complete.

We observe network formation in many situations, from blood veines formation to pattern formation of vascular network in plants' leaves. In the latter, we know that Auxin, a phytohormone, is sufficient to induce differenciation of cells into vascular form. Auxin is produced in apical areas of shoots, and is actively pumped to the roots. It is known that the flaw of Auxin is due to polarity and position of transportation proteins in the plasmic membrane of cells.

But we can observe loops in the network of leaves, which is a paradox in polarity of cells. At one point of the loop one cell must be at egal distance in the two directions of the roots. So it can not be polarized. Until now, several models of leaves pattern formation gave patterns similar to primary veines, but no models dealt with secondary or tertiary veines formation, or even with loops.

We investigate a model taking in account the polarity of cells, according to concentration of transportation proteins in cells membran, and the differenciation into vascular cells. We think the position of Auxin production sites or alternatively changing position could generate the same loops as observed in natural leaves and may be sufficient to product secondary and tertiary networks.