Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion

Figures & data

Table 1.  Observed convergence rates, j denotes the refinement step, h denotes mesh size, n denotes the number of elements, and m denotes the number of nodes.

j	h	n	m	||e s ||ℓ^2(0, T;L ^2(Ω))	||e 1||ℓ^2(0, T;L ^2(Ω))	||e 2||ℓ^2(0, T;L ^2(Ω))
1	h 0	8	9	1.75922909516429	1.11674591720160	1.10261347576481
2		32	25	0.45065854583694	0.27681838124441	0.26859338773873
3		128	81	0.11359187391040	0.06890292747305	0.06650989349128
4		512	289	0.02840019069468	0.01715510756730	0.01652957697302
5		2048	1089	0.00702348061909	0.00423712620012	0.00408005466671
6		8192	4225	0.00167301388341	0.00100896077326	9.71368531625e−04
7		32,768	16,641	3.3485e−004	2.0196e−004	1.9443e−004

Table 2.  Observed convergence rates, j denotes the refinement step, k denotes the time step, n denotes the number of elements, m denotes the number of time steps, is the error computed at the refinement step j, and gives the ratio of the error computed at two consecutive refinement steps.

j	k	n	m	||e s ||ℓ^2(0, T;L ^2(Ω))		||e 1||ℓ^2(0, T;L ^2(Ω))		||e 2||ℓ^2(0, T;L ^2(Ω))
1	k 0	25	50	0.6279	–	0.5780	–	0.4902	–
2		25	100	0.5305	1.18	0.4671	1.23	0.4006	1.22
3		25	200	0.4161	1.27	0.3556	1.31	0.3058	1.31
4		25	400	0.3078	1.35	0.2638	1.34	0.2294	1.33
5		25	800	0.2123	1.45	0.1857	1.42	0.1687	1.35
6		25	1600	0.1319	1.61	0.1169	1.59	0.1103	1.53
7		25	3200	0.0630	2.09	0.0563	2.07	0.0543	2.03

Table 3.  List of conditions on reproduction numbers and invasion numbers and potential dynamical outcomes of the competition of the strains. Last column gives the example where the case is numerically illustrated. Case 6 could not be numerically illustrated, possibly because it does not occur in system (1).

Case	Strain 1	Strain two	Expected outcome	Comment
1			I 1→0, I 2→0	Theorem 3.3
2			I 1→0, I 2 persists	Proposition 7.1, Example 2
3			I 1 persists, I 2→0	Proposition 7.1, Example 3
4	,	,	I 1→0, I 2 persists	See Example 4
5	,	,	I 1 persists, I 2→0	See Example 5
6	,	,	init. cond. dep. outcome	–
7	,	,	coexistence; I 1, I 2 persist	See Example 6

Figure 1. Initial values.

Figure 2. Total densities of susceptible individuals, infected individuals with strain one and two at time t.

Figure 3. Total densities of susceptible individuals, infected individuals with strain one and two at time t.

Figure 4. Total densities of susceptible individuals, infected individuals with strain one and two at time t.

Figure 5. Total densities of susceptible individuals, infected individuals with strain one and two at time t.

Figure 6. Total densities of susceptible individuals, infected individuals with strain one and two at time t.

Figure 7. Coexistence equilibrium.