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Let $e$ and $q$ be fixed co-prime integers satisfying $1\lt e\lt q$. Let $\mathscr {C}$ be a certain family of deformations of the curve $y^e=x^q$. That family is called the $(e,q)$-curve and is one of the types of curves called plane telescopic curves. Let $\varDelta$ be the discriminant of $\mathscr {C}$. Following pioneering work by Buchstaber and Leykin (BL), we determine the canonical basis $\{ L_j \}$ of the space of derivations tangent to the variety $\varDelta =0$ and describe their specific properties. Such a set $\{ L_j \}$ gives rise to a system of linear partial differential equations (heat equations) satisfied by the function $\sigma (u)$ associated with $\mathscr {C}$, and eventually gives its explicit power series expansion. This is a natural generalisation of Weierstrass’ result on his sigma function. We attempt to give an accessible description of various aspects of the BL theory. Especially, the text contains detailed proofs for several useful formulae and known facts since we know of no works which include their proofs.

Background

Collaborative care can support the treatment of depression in people with long-term conditions, but long-term benefits and costs are unknown.

Aims

To explore the long-term (24-month) effectiveness and cost-effectiveness of collaborative care in people with mental-physical multimorbidity.

Method

A cluster randomised trial compared collaborative care (integrated physical and mental healthcare) with usual care for depression alongside diabetes and/or coronary heart disease. Depression symptoms were measured by the symptom checklist-depression scale (SCL-D13). The economic evaluation was from the perspective of the English National Health Service.

Results

191 participants were allocated to collaborative care and 196 to usual care. At 24 months, the mean SCL-D13 score was 0.27 (95% CI, −0.48 to −0.06) lower in the collaborative care group alongside a gain of 0.14 (95% CI, 0.06-0.21) quality-adjusted life-years (QALYs). The cost per QALY gained was £13 069.

Conclusions

In the long term, collaborative care reduces depression and is potentially cost-effective at internationally accepted willingness-to-pay thresholds.

Declaration of interest

None.

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Theory of heat equations for sigma functions

Long-term clinical and cost-effectiveness of collaborative care (versus usual care) for people with mental–physical multimorbidity: cluster-randomised trial

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2 results

Theory of heat equations for sigma functions

Long-term clinical and cost-effectiveness of collaborative care (versus usual care) for people with mental–physical multimorbidity: cluster-randomised trial