Abstract
Approximating the time to extinction of infection is an important problem in infection modelling. A variety of different approaches have been proposed in the literature. We study the performance of a number of such methods, and characterise their performance in terms of simplicity, accuracy, and generality. To this end, we consider first the classic stochastic susceptible-infected-susceptible (SIS) model, and then a multi-dimensional generalisation of this which allows for Erlang distributed infectious periods. We find that (i) for a below-threshold infection initiated by a small number of infected individuals, approximation via a linear branching process works well; (ii) for an above-threshold infection initiated at endemic equilibrium, methods from Hamiltonian statistical mechanics yield correct asymptotic behaviour as population size becomes large; (iii) the widely-used Ornstein-Uhlenbeck diffusion approximation gives a very poor approximation, but may retain some value for qualitative comparisons in certain cases; (iv) a more detailed diffusion approximation can give good numerical approximation in certain circumstances, but does not provide correct large population asymptotic behaviour, and cannot be relied upon without some form of external validation (eg simulation studies).
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Acknowledgments
Elliott Tjia was supported by a studentship from the Engineering and Physical Sciences Research Council. The authors would like to thank Bernd Schroers, Robert Weston and Des Johnston for helpful discussions regarding the Hamiltonian approach.
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Appendix
Appendix
Freefem++ code (Freefem++ 2016) to numerically solve the Kolmogorov backward Eq. 22, in the case k = 2.
// set parameter values
real gamma= 1., beta= 2.0, n = 260., epsilon= 0.5;
// define the boundary
border B1a(t = 0,epsilon)x=t; y=epsilon;
border B1b(t = 0,epsilon)x=epsilon; y=epsilon-t;
border B2(t=epsilon,n)x=t; y = 0;}
border B3(t = 0,n)x=n-t; y=t;}
border B4(t = 0,n-epsilon)x = 0; y=n-t;}
// define mesh
mesh Th = buildmesh (B1a(10)+B1b(10)+B2(100)+B3(100)+B4(100));
// define finite element space
fespace Vh(Th,P2);
Vh tau,w;
// solve PDE
solve Backward (tau,w,solver=LU) =
int2d(Th) (-((beta/(2⋆n))⋆(x+y)⋆(n-x-y)+gamma⋆x)⋆dx(tau)⋆dx(w)
-gamma⋆(x+y)⋆dy(tau)⋆dy(w)
+gamma⋆x⋆dy(tau)⋆dx(w)
+gamma⋆x⋆dx(tau)⋆dy(w))
+int2d(Th)(2⋆gamma⋆(x-y)⋆w⋆dy(tau)
+((beta/n)⋆(x+y)⋆(n-x-y)-2⋆gamma⋆x-(beta/(2⋆n))⋆(n-2⋆x-2⋆y)
-gamma)⋆w⋆dx(tau))
-int2d(Th)(-w)
+ on(B1a,tau= 0) + on(B1b,tau= 0) ;
// Output results, for processing in Matlab
{ ofstream ff("SISk.txt");
for (int i = 0;i<Th.nt;i++)
{ for (int j = 0; j < 3; j++) ff<<Th[i][j].x << " "<< Th[i][j]
.y<< " "<<tau[][Vh(i,j)]<<endl;
ff<<Th[i][j].x << " "<< Th[i][j].y<< " "<<tau[][Vh(i,j)]<<endl;
ff<<Th[i][0].x << " "<< Th[i][0].y<< " "<<tau[][Vh(i,0)]<<"
∖n∖n∖n";
}
}
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Clancy, D., Tjia, E. Approximating Time to Extinction for Endemic Infection Models. Methodol Comput Appl Probab 20, 1043–1067 (2018). https://doi.org/10.1007/s11009-018-9621-8
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DOI: https://doi.org/10.1007/s11009-018-9621-8