= Solution
The third fit is a <negative binomial regression>. The Poisson model is obtained at the boundary where the negative-binomial overdispersion tends to zero, so the <likelihood-ratio test> has the asymptotic null distribution $\tfrac12\chi_0^2+\tfrac12\chi_1^2$. Since model3 has one additional parameter,
$$
D=2(\ell_3-\ell_1)
=\operatorname{AIC}_1-\operatorname{AIC}_3+2.
$$
The supplied output gives $\mathbb P(\chi_1^2\geq D)=0.04608555$, and hence the boundary-corrected p-value is
$$
p=\frac12(0.04608555)=0.023042775.
$$
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