**A new language for complex ODE models**

Michael Dunlavey(1), Shuhua Hu(1), Robert Leary(1)

(1) Certara/Pharsight Corp.

**Objectives:** Complex models such as HIV/HCV/HBV, and possibly some diabetes models as in the DDMoRe model repository[1], when represented as differential equations, tend to dramatically increase in length and width as modifications, such as mutations or treatments are made, even though the modifications are easily foreseen. The differential equations tend to have many repeated terms that are the same except for a sign, or almost the same as others. At the same time, the functional purpose of each term is often hard to discern. This weighs against the verifiability and modifiability of such models. The objective is to find a surface representation of such models that reduces these problems.

**Methods:** A prototype language is given, designed to allow easy extension of a model to include additional treatments and compartments. It allows easy extension along multiple dimensions, and it allows separation of independent model sections such as elimination, infection, activation, etc. The model specifies "dimensions", where a typical dimension could be active vs. resting, applied to cells of different types. It specifies "states" representing compartments, but they can be vectors indexed by one or more dimensions. It specifies "parameters" which can be vectors indexed by one or more dimensions. It specifies various types of flows, such as 0th-order, 1st-order, and 2nd-order. These flows can contain "i" and/or "j" in a state or parameter index. Acting as universal quantifiers, they are applied across multiple states or parameters. For example "decayrate T(i,j) del(i)" can indicate that T cells of all activities and infection states decay at the rate for the corresponding activity. If the character "*" appears in a state or parameter index, it indicates summation. For example if "T(active,*)" appears, it indicates the sum of active T cells over all infection states. The term "V(*)-V(noninf)" can represent the sum of the infective virus over all mutations (total minus noninfective virus).

**Results:** An HIV model given in Banks, et al [2] is encoded in the language. Then the model is modified to include a mutation, following Hu[3], and shown to simulate appropriately. The modification of the model is shown to require minimal ammendment to the original model statement.

**Conclusions:** A language is demonstrated that shows, in the context of an HIV model, that modifications such as addition of a virus mutation can be made with minimal coding effort and high verifiability.

**References:**

[1] DDMoRe Model Repository, http://repository.ddmore.eu/models

[2] Banks, H.T., Hu, S., Thompson, W.C. Modeling and inverse problems in the presence of uncertainty, CRC Press, 2014

[3] Hu, S., personal communication