83 problems

Consider a two-type SIS model in which the two species have general, possibly unequal, death rates and compete for space. For each species, define its birth/death ratio as its birt…

Modified reproduction-number conjecture. The modified reproduction number is

Limiting proportion conjecture. Given that the first and second epidemics are both large, $n^{-1}Z_n^{(2)}$ converges in probability to $z$ as $n\to\infty$.

Equality conjecture. The two critical densities coincide:

Network-robustness conjecture. The results are robust under the choice of network: other synthetic network models exhibit a subset of the same phases. In particular, non-spatial ne…

Shape theorem conjecture. For GIRGs, a shape theorem holds.

Let $\mathcal X$ be the state space of the network SIRS epidemic model, let $y$ denote the vector of infected-population variables, and suppose that an endemic equilibrium…

Consider the SIR-$\rho$ model on an Erdős–Rényi graph $G(n,\mu/n)$, with final epidemic size $\Lambda_n$, critical infection rate, and parameters $\rho$, $\gamma$, $\alpha$, and…

Consider the evoSIR model on an Erdős–Rényi random graph $G(n,\mu/n)$, with final epidemic size denoted by $\Lambda_n$. A discontinuous phase transition means that, as…

A travel ban is considered in a two-community epidemic model with infection spreading over a contact network and individuals traveling between communities. In the regime of very sp…

Uniform maximal-proportion conjecture. We conjecture that $y_{\max}(k)$ vanishes as $k$ goes to infinity.

Poisson maximal-proportion conjecture. We conjecture that $y_{\max}(\lambda)$ reaches its maximum for some $\lambda>0$.

Consider an epidemic model with two group structures that are partly overlapping, and let the final size denote the final proportion of the population infected. The overlap-monoton…

Let $(n_{1a}^*,n_{1i}^*,n_3^*)$ be an asymptotically stable equilibrium of the three-dimensional virus system, and let…

Assume that $r\kappa\mu_1\neq v\sigma$ and that the $(\operatorname{Coex}_{2,3})$-condition holds, which in particular implies $mv>r+v$. Let $(n_{1a}^*,n_{1i}^*,n_3^*)$ and…

Let $\boldsymbol{x}$ be the coordinatewise positive coexistence equilibrium arising when $\lambda_2>\lambda_1$ and both host types cannot invade one another while coexisting with t…

Convergence conjecture. Alternative (ii) of Theorem 2.1 holds: $\|S(t,\cdot)-\tilde{S}\|_{\infty}\to0$ and $\|I(t,\cdot)-\tilde{I}\|_{\infty}\to0$ as $t\to\infty$. The proposition…

Threshold conjecture. For all such $\gamma$ and $\nu$: if $\mathrm{R}_{0}(\gamma,\nu)<1$, then the $(\gamma,\nu)$-EpiSLFV process goes extinct; if $\mathrm{R}_{0}(\gamma,\nu)>1$, t…

Let $(\gamma,\nu)$-EpiSLFV be the epidemic process, and let (SC1) denote transient local survival, (SC2) permanent local survival, (SC3) transient global survival, and (SC4) perman…

Advection-induced asymmetry conjecture. The infection spreads not only due to diffusion but also due to advection, leading to asymmetric spatial distributions. The imaginary part o…

Consider the epidemic models labeled (1.3)–(1.7), ordered according to the number of nonlocal reaction terms they contain. Nonlocal-reaction spreading difficulty conjecture. The mo…

Consider the multi-species SLIR system for $V$ pathogens and $P$ species, with pathogens indexed by $a=1,dots,k$ satisfying $\mathcal{R}_{0a}<1$ and pathogens indexed by…

Finite-moment conjecture. For every non-constant solution $(\vec{S},\vec{I})(t)$, the incidence $g(t)$ has finite mean time and finite standard deviation.

Divergence conjecture. The time-integrated infected-vector population should diverge as the principal eigenvalue approaches zero from the positive side:

Continuity conjecture. The harvest should be continuous with respect to $\lambda_1(\mathcal{L}_{V_s})$, so that