Towards a formal model of addiction in individuals

We aim to create an explanatory formal model for addiction. We deem earlier attempts to create this type of model for addiction too complex and thus we propose to use just one ordinary differential equation: dN/dt = r * N * (1 - N/K) - (B * N^2) / (A^2 + N^2) This equation has been studied extensively since originally proposed by Ludwig et al. (1978) to model the outbreak of spruce budworms. From only the first term it would follow that N grows with rate r until limit K is reached. However, the second term controls the growth of N. The larger B in the second term the more the growth of N is controlled, with maximum control of the growth at A. How these parameters are interpreted in addiction depends on the specific type of addiction. In general, N can be thought to represent the consumption of an addictive substance or the frequency with which addictive behavior occurs. The r can be interpreted as the rate at which consumption leads to more consumption, which could for example be influenced by brain processes and peer pressure. B could represent the upper limit of self-control, which is reached if consumption gets so high that behavioral control is lost. 1/A can be thought of how fast self-control starts to influence behavior, which can for example be influenced by the social environment and beliefs about the consequences of the addictive behavior. The equilibrium states of the model we propose can be described by a cusp catastrophe model. In the cusp, there are two stable states, which could correspond to problematic or non-problematic behavior in terms of addiction. The behavior of the cusp catastrophe model can reproduce some of the important phenomena that are present in addiction. For one, quitting in addiction is hard, which corresponds to the hysteresis effect that we see in the cusp. Moreover quitting and relapsing are often sudden phase transitions just as can occur in a cusp catastrophe model. The cusp model also allows for more gradual changes which can be more appropriate for the initial transition to problematic behavior or substance use.