Persistent spatial patterns of interacting contagions

The spread of infectious diseases, rumors, fashions, and innovations are complex contagion processes, embedded in network and spatial contexts. While the studies in the former context are intensively expanded, the latter remains largely unexplored. In this paper, we investigate the pattern formation of an interacting contagion, where two infections, A and B, interact with each other and diffuse simultaneously in space. The contagion process for each follows the classical susceptible-infected-susceptible kinetics, and their interaction introduces a potential change in the secondary infection propensity compared to the baseline reproduction number $R_0$. We show that the nontrivial spatial infection patterns arise when the susceptible individuals move faster than the infected and the interaction between the two infections is neither too competitive nor too cooperative. Interestingly, the system exhibits pattern hysteresis phenomena, i.e., quite different parameter regions for patterns exist in the direction of increasing or decreasing $R_0$. Decreasing $R_0$ reveals remarkable enhancement in contagion prevalence, meaning that the eradication becomes difficult compared to the single-infection or coinfection without space. Linearization analysis supports our observations, and we have identified the required elements and dynamical mechanism, which suggests that these patterns are essentially Turing patterns. Our work thus reveals new complexities in interacting contagions and paves the way for further investigation because of its relevance to both biological and social contexts.

Figure 1

The model of interacting contagions. (a) Mean-field model (without space): Consider two infections, A and B, that circulate in a population. Four states are then possible for host individuals: susceptible S, partially infected A or B, and the coinfected state AB. In the contagion process, S becomes partially infected (A/B) with an initial infection rate $\alpha$ by contacting the infected; the partially infected individuals can be further be infected by the other infection to be doubly infected (AB) with the secondary infection rate ${\alpha }^{\prime }$. All infected individuals recover with rate $\beta$. (b) Spatially interacting contagions: When subpopulations are coupled through their spatial neighborhood, the diffusion captures the local mobility of individuals and thus also the infections they carried. Generally, the mobility of a given individual depends on its state; e.g., in epidemic spread, susceptible people statistically move faster than infected individuals, who might prefer to stay at home or in the hospital for recovery. Mathematically, this is captured by $D_S>D_{A,B,AB}$ within the RD framework described by Eq. (2).

Figure 2

Emergence of infection patterns in 1D space. Starting with a perturbed contagion-free state (random initial conditions), strongly and weakly infected regions segregate from each other as time goes by [(a), (b), (d), and (e)]. Note that, due to the symmetrical parameters for the two infections, the resulting patterns of $I_A$ and $I_B$ are also in symmetry as $t\to \infty$, even though their initial conditions are not. (c) The spatial heterogeneity $h\left(t\right)$ gradually increases and then tends to saturation thereafter. (f) Among all Fourier modes, there are some unstable ${\lambda }_{\max }^k>0$, which trigger the spatial instability, in line with the patterns shown here. Parameters: $R_0=2$, $C=1$, $D_S=10$, $D_I=1$.

Figure 3

The impact of contagion interactions. Upper row: The spatiotemporal evolution of overall density of infection A (${\rho }_{{}_A}=I_A+I_{AB}$) for $C=0.2,2.5,3$, which shows that strong competition, e.g., $C=0.2$ in (a) or strong cooperation like $C=3$ in (c) actually inhibits patterns. (b) A case with a relatively strong interaction ($C=2.5$) that takes a long time to develop patterns. Lower row: $h\left(t\right)$ for a couple of interaction $C$ indicates that patterns are most likely to happen in the case of $C\simeq 1$ [(d) and (e)]; any deviation to a smaller or larger value will delay or just fail to have the formation process. (f) Eigenvalue analysis shows that the pattern appears within $0.23<C_{\mathrm{instability}}<2.73$, where ${\lambda }_{\max }>0$ and the peak is around $C=1$, in line with the observations here. Parameters: $R_0=2$, $D_S=10$, $D_I=1$. Random initial conditions are used in (a)–(e).

Figure 4

The impact of baseline reproduction number. (a) ${\lambda }_{\max }$ versus $R_0$ for different $D_S$ by fixing $C=1$. (b) ${\lambda }_{\max }$ versus $R_0$ for different $C$ by fixing $D_S=10$. (c) Boundaries for a couple of $D_S$ in $R_0\text{−}C$ parameter space, where the left side allows for pattern (${\lambda }_{\max }>0$) and the right side corresponds to no pattern (${\lambda }_{\max }=0$). Similarly to the impact of the interaction strength $C$, there is also an upper threshold of $R_0$ for the pattern emergence, above which patterns disappear. Parameter: $D_I=1$.

Figure 5

The impact of mobilities. (a) ${\lambda }_{\max }$ shown in $D_S\text{−}C$ space, where the boundary line separates the regions with (${\lambda }_{\max }>0$) and without (${\lambda }_{\max }=0$) pattern formation. $D_I=1$ fixed. (b) Boundaries for a couple of $D_I$; the left sides are regions allowing patterns. Parameter: $R_0=2$.

Figure 6

Pattern hysteresis. The pattern hysteresis emerges when we increase the parameter $R_0$ (upper row) and then decrease it (lower row). Left column [(a) and (c)]: The average prevalence as a function of $R_0$ with and without (MF cases) space in both directions; the patterns present in the shaded region are indicated by a nonzero heterogeneity $h$. Right column [(b) and (d)]: The corresponding spatiotemporal evolution of patterns. The parameter $R_0$ is changed slowly enough (subplots at the top) to have stable patterns. Pattern hysteresis is defined by two sets of pattern regions [1–1.5 in (a) and (b) and 1.08–0.25 in (c) and (d)]. The lower panels show that the presence of patterns leads to a much more difficult eradication because $R_0^{e,p}<R_0^e$, close to zero, which is bad for containment. Parameter: $C=10$ with tiny conservative noise kept in the system (see the text).

Figure 7

Dynamical mechanism. (a) Density profiles arranged within the high (H) and low (L) infected density regions in the 1D domain. The dotted-dashed line is the MF value (i.e., without space) of S for reference. (b) The effect of reactions in the dynamics of S [i.e., $f_S$ in Eq. (2)]. The positive reaction contribution (source) is to increase the density of S, whereas the negative one (sink) is to increase the infected at the expense of S instead. (c) The scheme of dynamical flows between two neighboring regions: The net reaction within the H regions is from the susceptible S of a low density to the infected ${\rho }_{{}_{A,B}}$ of a high density to be even higher and is reversed in the L regions—an aggregation process (the thick vertical arrows). Diffusion is always from high-density regions to low-density regions (the curved arrows). In such a way, a dynamical loop is formed, maintaining stable patterns. Same settings as in Fig. 2, and profiles in (a) are plotted after 1000 time units.

Figure 8

Onset of pattern formation. Panels (a)–(d) show the occurrence of pattern formation triggered by a perturbation. The initial conditions of the evolution are from equilibrium state ($S^{*}=I_A^{*}=I_B^{*}=I_{AB}^{*}=1/4$, for the chosen parameters) except for the center site at the 1D domain with a length of 60 units shown here. Parameters: $R_0=2$, $C=1$, $D_S=10$, $D_I=1$.

Figure 9

Typical patterns in the 2D domain. (a) Spot pattern ($R_0=1.5$) and (b) strip pattern ($R_0=2.2$). Random initial conditions and 500 time units transient are used. Dark blue indicates weakly infected regions. Parameters: $C=1$, $D_S=10$.