Threshold analysis of multi-group SIR models with heterogeneous mixing

Abstract

We analyze the spread of a social interaction agent of finite duration of interest, such as a petition, behavioral trend, or opinion, through a heterogeneous population using a multi-group SIR model. By integrating the model equations, we obtain explicit final-size relations and identify threshold conditions that determine whether propagation can be sustained. For one-, two-, and three-group systems, we show how within-group reinforcement and cross-group influence shape the geometry of the no-propagation boundary. The results provide a clear geometric characterization of heterogeneous diffusion and offer practical guidance for designing interventions that either promote or inhibit spread.

1 Introduction

Understanding how ideas, behaviors, and collective actions spread and subsequently fade in networked societies is an interesting problem in applied mathematics and related fields [1-11]. These processes for the diffusion of an agent in a society are shaped by many factors, including social connectivity, exposure frequency, digital platform algorithms, and the lifetime of attention to a given topic. Among the various modeling approaches, compartmental flow models offer a tractable yet powerful way to capture aggregate spreading dynamics across diverse contexts.

Beyond epidemiology, many finite-interest social diffusion processes exhibit the same Susceptible-Infected-Removed (SIR) structure proposed by [12], one of the most widely used models in studying epidemics [13-17]. In this setting, the spreading element, referred to here as a social interaction agent, may represent a rumor, petition, product or an opinion. This interpretation allows epidemic-type models to be used for social diffusion processes. Individuals who have not yet engaged form a susceptible pool; those actively sharing or advocating play the infectious role; and those who have already acted (and thus do not re-engage) are effectively removed. Petition-signing campaigns (e.g., Change.org, Avaaz) are one illustrative example. In this context, susceptible individuals are those who have not yet encountered or signed the petition, infectious individuals are those who have signed and are actively sharing or promoting it, and removed individuals are those who have either signed and ceased to promote it or have decided not to sign at all. Limited-time product adoptions and short-horizon mobilizations can be interpreted in the same way. In the epidemiological terms, contacts with infectious individuals transmit the agent; after a finite infectious period, individuals move to a removed state with effective immunity. As the susceptible pool shrinks, transmission dies out. The same mechanism underlies the social processes just noted, which saturate as the pool of potential adopters diminishes. Beyond these finite-interest SIR-type processes, there is also a large literature on opinion diffusion driven by continuous-valued beliefs, notably the Friedkin-Johnsen and Hegselmann-Krause models, in which agents repeatedly average their opinions over a social network [18-20]. While these models do not enforce an absorbing removed state, they share with the SIR framework the idea of compartmental, network-mediated dynamics and provide a complementary perspective on how social influence and connectivity shape collective outcomes.

In addition to continuous opinion dynamics models, there is a substantial body of work on rumor and opinion spreading that focuses on threshold-type behavior [21-24]. In these models, individuals do not adopt or share an idea immediately, but only after reaching a certain level of exposure. This can lead to nonlinear spreading patterns and, in some cases, sharp transitions between situations where an idea spreads widely and those where it quickly dies out. These effects are known to depend on factors such as population heterogeneity, network structure, and differences in individual behavior. This type of behavior is closely related to the notion of thresholds in epidemic models, where a key parameter determines whether spreading can persist. The presence of heterogeneous thresholds in social systems therefore provides further motivation for using structured SIR-type models, which offer a natural way to represent both individual differences and their impact on overall spreading dynamics. In this work, we adopt a multi-group (age- or class-structured) SIR framework with heterogeneous mixing, commonly referred to as the age-structured SIR model [16, 25]. Two principal approaches are used in the literature to incorporate age or class structure into transmission models. The first is a continuous-age formulation, where each compartment depends on both age and time, leading to systems of partial differential equations [26-28]. Although rich, this formulation is often difficult to implement numerically. The second approach, which we adopt here, is a multi-group formulation that replicates the SIR compartments for each subgroup and allows inter-group interactions; it remains computationally tractable while capturing key heterogeneities such as differences in contact intensity or social behavior.

Multi-group SIR models therefore offer a natural framework for describing the spread of opinions, rumours, or fake news across heterogeneous societies. In this setting, susceptibility represents individuals’ readiness to adopt or share an agent (for example, a petition, narrative, or product), while the infectious state corresponds to a temporary phase of actively forwarding, commenting on, or promoting it. Different demographic or social groups, such as children, young adults, and older individuals, or professional, activist, and casual-user communities, often exhibit distinct levels of receptiveness, attention span, and influence. Younger users may adopt and retransmit quickly but also lose interest rapidly; adults may be more selective yet highly influential within professional or family networks; older individuals may be slower to adopt but more persistent once engaged. The multi-group structure captures these differences through group-specific transmission and removal rates, as well as cross-group contact intensities. As a result, the same SIR mechanism that describes age-structured epidemics can also reproduce how a rumour or fake news item may ignite in one subgroup, spill over into others through inter-group exposure, and eventually saturate or die out depending on the heterogeneous thresholds governing each community.

In epidemics, vaccination and contact-reduction measures (e.g., distancing, masking, travel restrictions) are the main instruments for curbing transmission [29-34]. In the classical homogeneously mixed SIR model, the key parameter is the basic reproduction number , defined as the expected number of secondary cases generated by a single infectious individual introduced in a fully susceptible population. This parameter partitions the parameter space into two regimes: below the threshold every surge dies out; above it a wave starts and grows. The present paper extends this threshold logic to heterogeneous (multi-group) settings and characterizes the boundary in parameter space that separates regions supporting sustained propagation from those in which any surge inevitably dies out. In the two-group parameterizations we consider, we prove that the threshold boundary is exactly a straight line in parameter space. The same linear threshold boundary occurs for the multi-group case under specific conditions described in Section 4, as hyperplanes in higher dimensional spaces. In particular, it has been shown that the linearity property of the boundary of the no-propagation region holds in the cases where within-group interaction strengths are equal across groups and cross group interaction strengths are proportional to a free single parameter. Furthermore, numerical simulations indicate that the boundary approximately preserves its hyperplane structure under perturbations of up to . The geometric description of the threshold boundary directly supports intervention design: starting from a point that sustains spread, one can cross into the no-spread region by selectively attenuating interactions (i.e., scaling entries of the contact matrix), with the preferred path dictated possibly by risk and cost. Knowing the analytic form of the boundary enables more targeted and efficient strategies for managing epidemics and, by analogy, for controlling the spread of information, promoting petition-signing campaigns or introducing a new product in a market.

To the best of our knowledge the investigation of a boundary for the propagation and no propagation regions in the parameter space has not been studied in the literature, in particular for control purposes of social interactions.

The article is organized as follows. Section 2 reviews the homogeneous SIR baseline and introduces the heterogeneous multi-group model. Section 3 derives final-size relations and threshold conditions via the next-generation matrix, including closed-form characterizations for two- and three-group specializations. Section 4 presents numerical studies that visualize threshold curves and surfaces, and no-propagation regions under representative scenarios. Section 5 discusses control strategies and Section 6 concludes with implications and possible extensions.

2 Epidemic models

Epidemic models define how contagion spreads through contacts between susceptible and infectious individuals. The classical Susceptible-Infected-Removed (SIR) system provides a baseline governed by the transmission rate and removal rate , whose ratio acts as a threshold value called the basic reproduction number: an outbreak decays for and grows for . In Section 2.1, we briefly recall the analytical structure and interpretation of the homogeneous SIR model to fix notation and highlight the role of as a control parameter; this baseline then motivates the heterogeneous, multi-group generalizations developed in Section 2.2.

2.1 The SIR model

We consider the classical SIR systemwith nonnegative state variables and positive parameters (transmission rate) and (removal/recovery rate). The model conserves total population as , so after normalization we may assume for all .

A simple threshold parameter governs the dynamics of this system. If , the infective class decreases monotonically and trajectories approach the disease-free state; if , incidence grows until the susceptible pool drops below , after which and the wave declines. Two control measures align with this threshold logic: vaccination (or any abrupt depletion of susceptibles) pushes below without changing the ratio and thus forces decay; by contrast, contact-reducing measures (e.g., distancing, masking, mobility limits) act by lowering and may move the system into the regime Peker-dobie et al. [32]. From the point of view of implementing control strategies, it is important to note that purely qualitative observations cannot, by themselves, distinguish the regimes versus. On the other hand, a single observation of past growth is decisive: if has been observed to increase at any time, then necessarily at that time; since , this implies . In other words, any historical increase in certifies a ratio , even if the system later transitions into decay once falls below .

A case with can be observed after an earlier phase with has generated infections in a fully susceptible population. Then, at some time when , the parameters change (e.g., due to interventions or a change in the agent) to a new value . Solving Equation 1 with the initial data , one finds that decreases and the epidemic dies out. Dividing by and integrating, one can obtain as a function of aswhere is the value of at some initial time . As , we have . For , at a time , with , is given by

Then if for , system parameters are changed to and , then with the initial conditions at , is expressed as

We rearrange the terms and rewrite aswhere . As , we have , and . Thus, the final values for the post-change phase satisfyfrom which we can solve

This equation is valid in the domain , and since we have . These relations imply a one-to-one correspondence between the final value of removed individuals and [35].

It is sometimes convenient to express in terms of initial values at a time . These values satisfy and therefore cannot be chosen arbitrarily. The special case corresponds to a trajectory with as . As a function of , behaves as follows: for it starts at 1 and is defined for all ; for , the value of remains greater than 1 and is defined for .

The parameters of the model proposed in the epidemiological setting, admit an interpretation in the context of social interactions. In particular, in the age of extreme digital connectivity, ideas and rumors spread uncontrollably with a limited lifespan. In our setting, the epidemiological “virus” has a natural analogue in social systems: we call it the agent; the item that spreads through interpersonal contacts (for example, a new product, a fashion trend, an ideology, or a petition campaign). Framed in the SIR model, an individual who adopts the agent enters a finite active period, analogous to the infective period, during which they communicate the agent to others through contacts. The active period is determined by the dynamics of spread rather than by symptoms, while the permanent effects of the agent may differ across groups of individuals. When the active period ends, individuals cease to promote the agent, though they may continue to be affected by it (for instance, continuing to use the product) without further contributing to its spread.

The parameters of the epidemiological model have the following interpretations in the context of social interactions. The transmission parameter is taken to comprise two components—the virulence of the virus and the contact rate , both shaping the evolution of the process. The contact rate has the same interpretation in the context o social interactions, while influential strength of the agent in various groups is the analogue of the virulence of the virus. The total number of individuals affected provides one measure of severity, yet severity may vary markedly across groups. For instance, H1N1 appears to have had milder effects in some older cohorts (likely due to partial immunity from earlier, related strains [36-38], whereas COVID-19 caused disproportionately severe outcomes among older adults, often due to complications [39, 40]. Such heterogeneity can be represented by a group-specific death (mortality) rate for group , allowing the model to reflect age-/group-dependent fatality risks alongside overall incidence. SIR models with heterogeneous populations are used to capture these subtleties. In the next subsection, we concentrate on the SIR model for multiple age classes.

2.2 SIR model for multiple age classes

In this section, we consider an age-structured SIR model under heterogeneous mixing. In a society with heterogeneous mixing, the rate at which individuals in one group interact with those in another generally differs across pairs of groups [25]. The multi-group formulation approach replicates the Susceptible-Infected-Removed (and, if included, Vaccinated) compartments for each age class and allows interactions between classes. Although this increases the number of equations, it remains numerically tractable and is widely used in practice. A simple, yet typical, example of heterogeneous mixing is a population consisting of children, young adults and elderly people. This coarse subdivision was reflected during the COVID-19 pandemic: elderly people were the most vulnerable group requiring protection, whereas general confinement measures disrupted schooling for children and curtailed economic activity among young adults.

Differences in contact rates between age groups influence the shape of the epidemic curve and can also affect the optimality of intervention strategies [25]. The parameter , which characterizes the force of infection in the SIR model, is a product of the virulence of the agent and the contact rate between susceptible and infected individuals.

In heterogeneous models, it is necessary to represent the contributions of the virulence of the infecting agent and the contacts among individuals by distinct parameters. The virulence denoted by may differ across groups; furthermore, the probability of transmission, denoted by , need not be the same for all groups. The resulting system of differential equations is expressed as follows.

Here, the contact effect between the groups and is represented by , the mean number of contacts made by an individual in group with individuals in group . In general, . However, the total number of mutual contacts between groups and must satisfy the following reciprocity relation

Hence,

The differential equations can be integrated as follows. In Equation 2, we replace by (from Equation 4) to get

Note that as S + I + R = 1, Equation 3 follows from Equation 2 and Equation 4.

Then, dividing the resulting equation by and integrating, we obtainwhere is an integration constant. If the initial value of is , then and are zero initially, and

At the final steady state, and . Thus, we obtain the following algebraic relation among the final values

Note that , for all , is a solution. The left hand side of the equation becomes undefined when ; therefore must lie within the range .

3 Analytical results: final-size relations and thresholds

This section develops the analytical structure of the multi-group opinion-spread model introduced in Section 2. Using the integrated form of the governing equations, we derive explicit relations that connect the long-term adoption levels in each subgroup to the interaction parameters governing within-group reinforcement, cross-group influence, and the duration of active participation. These relations generalize the familiar threshold behavior observed in single-group diffusion models, showing how heterogeneity across groups can either facilitate or suppress the spread of an opinionated agent. By examining one-, two-, and three-dimensional configurations of the interaction matrix, we identify parameter combinations that permit sustained propagation as well as those leading to decay. In several cases the boundary separating these regimes assumes a simple geometric form, allowing clear visualization of how reinforcement and cross-exposure jointly shape the eventual impact of the agent.

3.1 One-dimensional case

We recall the SIR system defined in (Equation 1) with the normalization condition . This system can be solved with initial conditions at corresponding to . The solution satisfies

Thus, initial conditions at some finite time are

The system admits two equilibrium points, (1,0,0) and , i.e., . The equilibrium value is obtained by solving the nonlinear equationor equivalently,

The solution of this nonlinear equation corresponds to the intersection of the curves and . The first function is monotonically increasing with slope equal to 1. Thus, if , the system has no nontrivial solution.

If the initial conditions at minus infinity are (1,0,0), (i.e., the initial conditions at any finite time cannot be specified freely), then for , the final value is zero. Otherwise, there exists a nonzero final value determined implicitly by the ratio .

The parameter , called the Basic Reproduction Number and denoted by , is the key threshold parameter of the 1-dimensional SIR model.

Let , where denotes the virulence of the infection agent and represents the contact rate. Then, the relation above can be interpreted as expressing in terms of the contact rate. The sensitivity of to variations in the contact rate can therefore be analyzed directly. In particular, one can determine threshold values for the contact rate that would eliminate the possibility of an epidemic.

As seen in Figure 1, the function is monotonically increasing on , satisfies , and diverges as . Hence a nontrivial final size exists if and only if ; when , the only admissible final value is .

FIGURE 1

Note that as , the no-propagation condition determines a region in the parameter space as the region in the first quadrant lying below the curve . In the present work we aim to generalize this interpretation of the basic reproduction number, as conditions determining no-propagation regions in the parameter spaces of the problems under consideration. In the multi-group setting, we will adopt the definition of as the largest eigenvalue of the next-generation matrix, as defined in [41, 42].

3.2 Two-dimensional case

We consider a society consisting of two homogeneous groups. Heterogeneity with respect to a given agent may arise (i) from inherent properties that differ across groups (for example, different removal/active-period rates), (ii) from contact opportunities (unequal mixing within- and between groups), or (iii) from group-specific effects of the agent (for example, relatively harmless in one group but severe in another). In social diffusion the analogues are different attention spans, different exposure patterns, and different behavioral responses (e.g., a game more addictive for children, certain ads more effective for younger people). Hence, rather than inhibiting or enhancing contacts uniformly, it is often preferable to direct controls selectively.

We consider the following two-group SIR system

In this setup, the parameters , and represent intrinsic properties of subpopulations, while the contact intensities can be considered as control parameters.

We compute the basic reproduction number , following the next-generation matrix (NGM) approach [41, 42]. The disease free equilibrium is defined byand the Jacobians of the new-infection terms and the other transitions at the disease free equilibrium are given by, respectively

Thus, the next-generation matrix isand the basic reproduction number is defined as where refers to the spectral radius of the matrix . In terms of the entries of the next-generation matrix, , the characteristic equation given byand the largest root of this equation is evaluated as

The boundary of the no-propagation region is obtained by replacing and rearranging terms to obtain

The boundary of the no-propagation region is therefore given byand the positivity requirements of the parameters,

The propagation threshold is ; parameter combinations with do not sustain spread, while those with do.

Two specializations make the geometry transparent and are useful for design. Consider the case where within-groups are of comparable strength in the sense that the diagonal entries of the next-generation matrix may be assumed to be equal to each other, i. e.,and the strengths of interactions among different groups are proportional, i. e., the off-diagonal entries of the next-generation matrix are

Thus the next-generation matrix has the formthe basic reproduction number is and the no-propagation region is the triangular region in the first quadrant of the plane , lying below the line

Remark. The formation of a linear threshold boundary is due to a property of the structure of the next-generation matrix, leading to the linearity of its eigenvalues. Namely, if a matrix can be written as a linear combination of constant matrices that are simultaneously diagonalizable, then its eigenvalues are also a linear combination of the eigenvalues of the components of the sum. In particular the linear threshold structure observed in the two-group case is not restricted to dimension two, but it is due to the selection of the interaction structure. Assuming that the next-generation matrix can be written in the form where represents a common within-group reinforcement level and is a fixed matrix defining the interaction pattern between groups. Then, the eigenvalues of are given by for . Thus, the threshold condition is linear in the parameters and , and the corresponding no-propagation boundary is a hyperplane in the related parameter space. The form above corresponds to a scenario in which all groups have the same level of within-group reinforcement, while the cross-group interaction structure is fixed and scaled by a common parameter. In this case, the threshold for propagation depends linearly on the contributions of within-group reinforcement and cross-group exposure. However, the linearity of the boundary is no longer observed in cases where within-group interactions are different and/or cross-group interactions can vary independently.

3.3 Three-dimensional case

Let us consider the following three-dimensional systemwhose next-generation matrix is

The characteristic equation for the next-generation matrix iswhere

Under the specialization , , , and , the problem reduces to finding the spectral radius of

The determination of the spectral radius, hence of the basic reproduction number reduces to finding the largest root of the characteristic equation, which is a third order polynomial. When is symmetric, we obtain the relatively simple formula given below. Note that and is the largest root of the following cubic characteristic equation

The discriminant of the equation aboveis always non-positive since the arithmetic mean is always greater than or equal to the geometric mean. Therefore, all the roots of this equation, one positive and two negative, are real given bywhere . Hence, .

Case 1 (Equal cross-terms): If the parameters and are all equal, then substituting in (Equation 6), we obtain as the boundary of the no propagation region.

Case 2 (Unequal cross-terms): If the parameters , and are not independent but are all multiples of a single parameter, say , then Equation 6 can be used to obtain the solution in the form , where is a constant computed from Equation 6, hence the boundary of the no propagation region is linear. For example, if , then the boundary of the no-propagation region is given by the line .

On the other hand, if the parameters are independent, then the boundary of the no-propagation region will be given by Equation 6 and it will be a hypersurface in three dimensional space as shown in Figure 2.

FIGURE 2

4 No-propagation regions

In the remainder of this section, we examine a sequence of canonical configurations of the matrix that correspond to progressively more complicated mixing patterns. By analyzing these forms, we isolate the fundamental mechanisms through which different types of heterogeneity influence whether an agent can propagate through interacting subgroups. Sections 4.1-4.3 therefore present no-propagation regions for increasingly heterogeneous parameter choices. In Section 4.1, we consider equal within- and cross-group interactions, leading to a linear threshold boundary. In Section 4.2, we introduce unequal cross-group influences that depend on a single independent parameter, hence the threshold boundary is still linear. Finally, in Section 4.3, we consider unequal within-group amplification, leading to a nonlinear threshold boundary. In each case, we compare the theoretical linear or curved threshold under parameter perturbations.

The selection of the parameter values in our simulations reflect relative strengths of the activities of various subgroups. The level of perturbations is limited to for clearer visibility of the boundaries.

4.1 Equal within- and cross-group interactions

In this first configuration we examine the case in which all groups have identical within-group reinforcement and exert equal influence on each other. This structure provides the simplest benchmark for visualizing the geometry of the no-propagation boundary.

For the two-group system, the mixing matrix takes the form . Its largest eigenvalue is , so the threshold condition reduces to the linear boundary . All points satisfying lie in the no-propagation region, while points above this line sustain spread. Figure 3 (left panel) visualizes this region: the shaded triangular domain corresponds exactly to and even under 10 perturbations the geometry remains aligned with the theoretical straight-line boundary.

FIGURE 3

For the three-group case, we consider the fully symmetric matrix . In this three-group system, the largest eigenvalue is , so the no-propagation boundary is again linear and given by . Figure 4 (right panel) confirms this: the shaded region matches the threshold line, and perturbations do not alter the essentially linear shape of the boundary.

FIGURE 4

4.2 Unequal cross-group interactions with equal within-group reinforcement

We next consider the case of different interaction strength between groups, while the within-group reinforcement remains the same across all groups. Thus, the diagonal entries of the matrix are equal, and all heterogeneity comes from the cross-group interaction terms. This setting reflects the fact that, in social propagation, influence between groups is often directional rather than balanced. Compared with the case in Section 4.1, the contribution of the off-diagonal terms changes the threshold condition. However, as long as these unequal cross-group interactions are proportional to a single independent parameter, they do not destroy the linear structure of the no-propagation boundary.

For the two-group system we consider , in which the influence from group 2 to group 1 is twice as strong as the influence in the opposite direction while both groups share the same within-group parameter . The largest eigenvalue of this matrix is , so the threshold is the straight line . Figure 4 illustrates the no-propagation region satisfying . Even with 10 perturbations applied to all parameters, the threshold remains aligned with the theoretical straight-line boundary.

A natural three-group analogue is shown in Figure 5, where again all diagonal entries are equal but two of the cross-group pathways are stronger than the others, yielding the matrix . Here, the second and third groups exert a stronger mutual influence compared to their influence on the first group .

FIGURE 5

A further three-group analogue is shown in Figure 6, where the diagonal entries remain equal but the cross-group influences differ across pairs. The resulting matrix has the form . The dominant eigenvalue is so the no-propagation boundary is again a straight line.

FIGURE 6

4.3 Unequal within-group amplification with equal cross-group interactions

In this final configuration, we examine cases in which within-group reinforcement differs across groups while cross-group interactions remain identical. Such a structure captures situations where one subgroup is intrinsically more active, responsive, or persistent than the others, even though all groups interact with one another in an identical way. Socially, this corresponds to a situation in which some groups reinforce the agent more strongly within themselves than others, while the channels of interaction between groups remain unchanged. In this case, propagation is no longer determined by a simple additive balance between within-group and cross-group effects. Rather, the groups with stronger internal reinforcement become more decisive, so that even small changes in those groups may have a disproportionately large effect on whether propagation is sustained.

For two groups, we consider the matrix where the second group reinforces itself at twice the level of the first group. In this context, as long as cross-group interactions depend on a single parameter, the nonlinearity of the threshold boundary is primarily caused by the inequality of the within-group interactions, thus to simplify the presentation we consider equal cross-group interaction parameter . Because the diagonal entries differ, the dominant eigenvalue becomes which depends nonlinearly on the parameters and . Therefore, the boundary is no longer a straight line but a smooth nonlinear curve, as seen in Figure 7.

FIGURE 7

A three-group analogue is presented in Figure 8, where the matrix is . The dominant eigenvalue depends nonlinearly on the parameters and , and hence, the no-propagation boundary is curved. Figure 8 shows that this curvature becomes more pronounced as diagonal asymmetry increases: the stronger the internal amplification of the third group, the steeper and more compressed the feasible region becomes. As in the two-group case, 10 perturbations do not alter the nonlinear geometry of the boundary.

FIGURE 8

5 Implementation of control strategies

In this section we illustrate the analytical results for the control of the spread of an agent in the two-group setting. The starting point is Equation 5 that relates the final sizes of removed individuals in each group to the system parameters. In Section 4, we studied this equation with the aim of determining the regions in the parameter space that would sustain or inhibit the spread of an infective agent. In this section we study the problem of determining the parameters that would result in prescribed (non-zero) final values of removed individuals for each group.

Definingthe equations relating final sizes for two subgroups are written as,

Recall that , hence the system above actually consists of two equations for three unknowns, namely, , and .

For prescribed values of , this system is linear in the parameters , and , and can be easily solved, yet meaningful solutions should satisfy positivity constraints, hence all targeted final values for may not be achievable for a given set of fixed parameters. We illustrate the implementation of control strategies for a typical application, where a small group of influencers aim to spread an opinion within a larger but relatively less active group, to reach given values of and , by determining . As the group 1 is considered as the control group, and are the control parameters and acts as a fixed parameter. Hence

We may assume that , because although there are few influencers, they are more likely to affect members of group 2. Assuming equal infection periods and equal virulence we have a simpler systemthat can also be expressed as

Hence the control parameters are

We can solve this system.

Note that the first terms are positive, whereas the seconds terms are negative. For given choices of and , positivity of overall mean contact rates is determined by . This example illustrates that the achievability of target values for may depend on fixed parameters of the system.

This linear structure has two practical consequences. First, it enables forward design of interventions: given desired final levels and , one can determine which entries of the contact matrix must be increased or decreased to achieve these outcomes. Second, it allows backward interpretation of observed data: measured final sizes give direct information about the underlying interaction structure, enabling estimation of which contact pathways are most influential. These insights support a range of applications, from amplifying adoption in balanced systems to suppressing spread selectively when one subgroup is more sensitive or at higher risk.

6 Conclusion

In this paper we analyzed multi-group SIR models with heterogeneous mixing as a flexible framework for studying finite-interest diffusion processes. Working with a next-generation matrix formulation, we showed how the basic reproduction number depends on group-specific transmission and removal rates together with asymmetric contact patterns. For a range of two- and three-group parameterizations we proved that the threshold boundary reduces to a simple linear condition in the relevant parameters, and numerical computations indicate that in higher-dimensional settings the invasion boundary is approximated by a hyperplane. These structural results provide a transparent way to understand how age or group heterogeneity shapes the transition between non-propagation and large-scale spread.

Beyond their epidemiological origins, multi-group SIR models are well suited to describe the diffusion of opinions, rumours and fake news across heterogeneous societies. In such applications, susceptibility quantifies individuals’ readiness to engage with or share an item (for example, a petition or narrative), while the infectious state corresponds to a transient phase of active forwarding, commenting or promoting. The multi-group structure allows different demographic or social groups to have distinct engagement and disengagement rates, as well as asymmetric cross-group influence, and the threshold analysis clarifies how an agent may ignite in one subgroup, spill over into others and eventually saturate or die out as finite attention is exhausted.

Classical opinion-dynamics models such as those of Friedkin–Johnsen and Hegselmann–Krause emphasise the evolution of persistent, continuous-valued beliefs through repeated averaging on a social network. On the other hand, the multi-group SIR perspective highlights temporary activation driven by contact events and finite-interest lifetimes. These viewpoints are complementary rather than competing. A natural direction for future work is to combine them explicitly, for instance by coupling an SIR-type engagement layer to a slower opinion-revision mechanism. Such hybrid models could capture both the rapid spread of participation (signing, sharing, endorsing) and the gradual evolution of underlying attitudes, and may help to integrate epidemic-style threshold analysis with the rich phenomenology of opinion formation observed in social systems. As a final remark we recall that in this paper, we concentrated our efforts for the control of the spread of a social agent via the control of the contact rates of the most active groups in a heterogeneous society. Alternatively, one may consider the spread of counter information to reduce the interest in the current agent, as an analogue of vaccination. The reduction of the lifespan of a social agent is also a promising control mechanism with no epidemiological analogue.

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