Optimal voluntary and mandatory insect repellent usage and emigration strategies to control the chikungunya outbreak on Reunion Island

Optimal levels of voluntary insect repellent usage

We adopt a modeling approach of Bauch & Earn (2004) and Dorsett et al. (2016). The strategy of an individual is the proportion of the day r ∈ [0, 1] the individual is protected from mosquito bites; the protection is granted by insect repellent. We assume that the repellent provides complete protection from mosquito bites while it is active. Since mosquitoes cannot bites humans while they are protected by the insect repellent, the mosquito-to-human transmission coefficient β₁ becomes a function of r. If no protection is used (r = 0), then β₁(0) is at its base value ${\beta }_1^0$ (given in Table 1). If humans are protected at all times (r = 1), then mosquitoes cannot bite these humans at all, and hence they cannot infect humans: β₁(1) = 0. We therefore assume that the mosquito-to-human transmission coefficient is a linear function of r given by (7)${\displaystyle {\beta }_1={\beta }_1^0\left(1-r\right).}$

If all susceptible humans in the population adopt the same strategy r_pop, then the basic reproduction number becomes a function of r_pop by substituting the expression (7) for β₁ into (5). The graph of the basic reproduction number as a function of the population strategy r_pop is shown in Fig. 2. The herd immunity protection level r_HI is the population protection level that reaches the threshold R₀ = 1 for disease eradication.

We define the utility function (expected payoff) of a susceptible individual using strategy r in a population that adopted strategy r_pop as (8)${\displaystyle E\left(r,r_{\text{pop}}\right)=-\pi \left(r,r_{\text{pop}}\right)C_i-r{C}_p,}$ where C_i is the cost of infection, C_p is the cost of complete protection through insect repellent, and π(r, r_pop) is the probability of infection. The latter depends on the individual’s strategy r because it determines how often mosquitoes may bite the individual, and on the population strategy r_pop because it affects the prevalence of the disease (e.g., the number of infected mosquitoes). The outcome of a game does not change if the utility function is scaled, so we divide the right-hand side of (8) by C_i to obtain (9)${\displaystyle E\left(r,r_{\text{pop}}\right)=-\pi \left(r,r_{\text{pop}}\right)-rC,}$ where C = C_p∕C_i is the cost of complete protection relative to the cost of infection.

We next compute the probability of getting infected and becoming symptomatic as the transition probability from the susceptible compartment S to the symptomatic infectious compartment I. This probability is the product of the probability that a susceptible individual becomes exposed f₁∕(μ₁ + f₁) multiplied by the probability that an exposed individual becomes symptomatically infected ϕλ₁∕(μ₁ + ϕλ₁ + (1 − ϕ)λ₁) = ϕλ₁∕(μ₁ + λ₁): (10)${\displaystyle \pi \left(r,r_{\text{pop}}\right)=\left(\frac{f_1\left(r,r_{\text{pop}}\right)}{{\mu }_1+f_1\left(r,r_{\text{pop}}\right)}\right)\left(\frac{\varphi {\lambda }_1}{{\mu }_1+{\lambda }_1}\right),}$ where (11)${\displaystyle f_1\left(r,r_{\text{pop}}\right)={\beta }_1^0\left(1-r\right)\frac{Z^{\ast }}{N^{\ast }}}$ is the force of infection, which depends on the individual protection level r and on the population protection level r_pop. The individual protection level r determines the rate at which mosquitoes bite the individual ${\beta }_1^0\left(1-r\right)$. The population protection level r_pop affects the prevalence of the disease in the population; it determines the size of the compartment Z^∗ via the substitution of the expression ${\beta }_1^0\left(1-r_{\text{pop}}\right)$ for β₁ into (6). In particular, Z^∗ and N^∗ do not depend on the individual protection level r.

To find the Nash equilibrium population protection level, we attempt to maximize the utility function (9) of a focal individual. Observe that (12)${\displaystyle f_1\left(r,r_{\text{pop}}\right)=\left(1-r\right)f_1\left(0,r_{\text{pop}}\right),}$ and hence (13)${\displaystyle \frac{\partial }{\partial r}{f}_1\left(r,r_{\text{pop}}\right)=-f_1\left(0,r_{\text{pop}}\right).}$ It follows that (14)${\displaystyle \frac{{\partial }^2}{\partial r^2}E\left(r,r_{\text{pop}}\right)=\frac{2{\mu }_1{f}_1{\left(0,r_{\text{pop}}\right)}^2}{{\left({\mu }_1+f_1\left(r,r_{\text{pop}}\right)\right)}^3}>0.}$ Consequently, the utility function is a convex function of r, and thus it attains its maximum value at one of the endpoints: r = 0 or r = 1.

This conclusion can be interpreted as follows. If the population repellent usage is sufficiently high, then the probability of getting infected is very low. A focal individual would rather bypass the potentially costly preventive measure and face the low morbidity risk instead. Hence individuals may improve their payoff by deviating from the population strategy (they should stop using repellent). On the other hand, if the population repellent usage is low, then the probability of getting infected is too high, and a focal individual should prefer to pay the cost of complete protection rather than face the high morbidity risk. In this case, individuals may also improve their payoff by deviating from the population strategy (they should use repellent 100% of the time).

So, if the population repellent usage is too high, then individuals would do better if they stop using repellent, and hence the population repellent usage will decrease. Conversely, if the population repellent usage is too low, then individuals would do better if they start using repellent 100% of the time, and hence the population repellent usage will increase. If the population repellent usage is “just right” (Nash equilibrium), then individuals cannot improve their payoffs by using repellent either less frequently or more frequently. We note that there is a presumption of the population-wide adoption of treatment rates in our model that is common to ESS-modeling; however, in situations such as (14), there is a potential implication that the population should in fact separate into distinct groups with different adoption rates. As it goes beyond the framework discussed here, we leave that investigation for future research.

The Nash equilibrium protection level of the population r_NE is thus a solution to the equation (15)${\displaystyle E\left(0,r_{\text{NE}}\right)=E\left(1,r_{\text{NE}}\right)}$ or (16)${\displaystyle \frac{f_1\left(0,r_{\text{NE}}\right)}{{\mu }_1+f_1\left(0,r_{\text{NE}}\right)}\frac{\varphi {\lambda }_1}{{\mu }_1+{\lambda }_1}=C.}$ The graph of the optimal (Nash equilibrium) repellent usage as a function of the relative cost of protection C is shown in Fig. 3A. The optimal repellent usage r_NE reaches the herd immunity r_HI level only when the cost of the protective measure relative to the cost of chikungunya infection is negligible (i.e., zero mathematically). The optimal repellent usage remains very close to the herd immunity level for a range of values of the relative cost C, and then drops off sharply. Once the relative cost of protection becomes too large (C_max), then everyone stops using insect repellent because its high cost forces individuals to prefer to risk the cost of infection.

Optimal levels of mandatory insect repellent usage

We now consider a scenario where an organization (e.g., the government) enforces the use of insect repellent in the population to fight chikungunya. The organization must balance the cost of prevention of the disease in the population and the cost of treatment of symptomatically infected individuals. On the one hand, every individual who utilizes insect repellent 100% of the time results in a cost C_p (same as the cost of voluntary complete protection). On the other hand, every symptomatically infected individual results in a cost C_i.

The goal of the mandating organization is to find the repellent usage level for the population r_pop ∈ [0, 1] so that the expected payoff (negative of the total cost) (17)${\displaystyle E\left(r_{\text{pop}}\right)=-C_i{I}^{\ast }-r_{\text{pop}}{C}_p{S}^{\ast }}$ is maximal. (Note that the equilibrium values I^∗ and S^∗ depend on r_pop.) Here we analyze the case where the mandating organization addresses mosquito-to-human transmission by advising susceptible individuals to spray themselves with insect repellent. One may also consider an alternative scenario where the infectious individuals are using insect repellent to reduce the human-to-mosquito transmission.

As before, we scale the payoff function and consider (18)${\displaystyle E\left(r_{\text{pop}}\right)=-I^{\ast }-r_{\text{pop}}C{S}^{\ast },}$ where C = C_p∕C_i is the relative cost of protection. The graphs of this function for different values of C are shown in Fig. 4. There are two possible outcomes: (1) the susceptible individuals should adopt the repellent usage level equal to that of the herd immunity threshold r_HI, leading to the eradication of the disease; or (2) no insect repellent should be used, and it is more cost-effective to treat symptomatically infected individuals only. The first outcome occurs for sufficiently low values of C (less than 0.00024), and the second outcome occurs for greater values of C (greater than 0.00024); the threshold value of C separating the two outcomes was found numerically.

Optimal levels of voluntary emigration

We are going to operate under the assumption that the chikungunya outbreak on Reunion Island did not affect the neighboring island of Mauritius Island, located 140 miles to the north-east of Reunion. Susceptible individuals—and only susceptible individuals, perhaps identified through a screening or quarantine procedure—may elect to emigrate from Reunion to Mauritius to protect themselves from the outbreak. To model the emigration as a potential personal protection measure against chikungunya, we allow residents of Reunion Island to leave the susceptible compartment of the epidemiological model at an emigration rate ω. This modification to (1) replaces the first equation describing the change in the susceptible population with (19)${\displaystyle \frac{\mathrm{d}S}{\mathrm{d}t}={\Lambda }_1-\frac{{\beta }_{1}SZ}{N}-{\mu }_{1}S-\omega S.}$ The DFE of the modified system is given by (20)${\displaystyle \left(S^0,E^0,I^0,I_a^0,R^0,X^0,Y^0,Z^0\right)=\left(\frac{{\Lambda }_1}{{\mu }_1+\omega },0,0,0,0,\frac{{\Lambda }_2}{{\mu }_2},0,0\right),}$ and the corresponding basic reproduction number of the disease is (21)${\displaystyle R_0=\frac{1}{{\mu }_2}\sqrt{\frac{{\Lambda }_2{\beta }_1{\beta }_2{\lambda }_1{\lambda }_2\left({\mu }_1+\omega \right)}{{\Lambda }_1\left(\gamma +{\mu }_1\right)\left({\lambda }_1+{\mu }_1\right)\left({\lambda }_2+{\mu }_2\right)}}.}$ The graph of the basic reproduction number as a function of the emigration rate ω is shown in Fig. 5. It is an increasing function of ω, and hence emigrating susceptible individuals paradoxically make it worse for the remaining susceptible population. When the fresh blood supply is reduced due to emigration, susceptible mosquitoes are more likely to prey upon infectious humans, increasing the disease prevalence in the vector population. Consequently, the remaining susceptible human population is at an increased risk of contracting the disease from a mosquito bite. It follows that, while being potentially beneficial to specific individuals, voluntary emigration may result in a tragedy-of-the-commons effect for the remaining islanders.

To further investigate the effect of voluntary emigration on the chikungunya epidemic and whether (selfish) susceptible individuals should emigrate, we compute the EE values of all compartments in the model with emigration: (22)${\displaystyle N^{\ast }=\frac{{\Lambda }_1{\mu }_1+\omega \left({\lambda }_1+{\mu }_1\right)E^{\ast }}{{\mu }_1\left({\mu }_1+\omega \right)},S^{\ast }=\frac{{\Lambda }_1-\left({\lambda }_1+{\mu }_1\right)E^{\ast }}{{\mu }_1+\omega },I^{\ast }=\frac{\varphi {\lambda }_1{E}^{\ast }}{\gamma +{\mu }_1},I_a^{\ast }=\frac{\left(1-\varphi \right){\lambda }_1{E}^{\ast }}{\gamma +{\mu }_1},R^{\ast }=\frac{\gamma {\lambda }_1{E}^{\ast }}{{\mu }_1\left(\gamma +{\mu }_1\right)},X^{\ast }=\frac{{\Lambda }_2\left(\gamma +{\mu }_1\right)\left({\Lambda }_1{\mu }_1+\omega \left({\lambda }_1+{\mu }_1\right)E^{\ast }\right)}{d+{\mu }_2\left(\gamma +{\mu }_1\right)\left({\Lambda }_1{\mu }_1+\omega \left({\lambda }_1+{\mu }_1\right)E^{\ast }\right)},Y^{\ast }=\frac{{\Lambda }_2{\beta }_2{\mu }_1{\lambda }_1\left({\mu }_1+\omega \right)E^{\ast }}{\left({\lambda }_2+{\mu }_2\right)\left[d+{\Lambda }_1{\mu }_1{\mu }_2\left(\gamma +{\mu }_1\right)+{\mu }_2\omega \left({\lambda }_1+{\mu }_1\right)\left(\gamma +{\mu }_1\right)E^{\ast }\right]},\text{and}{Z}^{\ast }=\frac{{\Lambda }_2{\beta }_2{\mu }_1{\lambda }_1{\lambda }_2\left({\mu }_1+\omega \right)E^{\ast }}{{\mu }_2\left({\lambda }_2+{\mu }_2\right)\left[d+{\Lambda }_1{\mu }_1{\mu }_2\left(\gamma +{\mu }_1\right)+{\mu }_2\omega \left({\lambda }_1+{\mu }_1\right)\left(\gamma +{\mu }_1\right)E^{\ast }\right]},}$ where (23)${\displaystyle d={\beta }_2{\lambda }_1{\mu }_1\left({\mu }_1+\omega \right)E^{\ast },}$ and E^∗ is the solution to the quadratic equation (24)${\displaystyle a{E}^2+bE+c=0}$ with coefficients (25)${\displaystyle a=-{\mu }_2\omega {\left({\lambda }_1+{\mu }_1\right)}^2\left({\lambda }_2+{\mu }_2\right)\left[{\beta }_2{\mu }_1{\lambda }_1\left({\mu }_1+\omega \right)+{\mu }_2\omega \left({\lambda }_1+{\mu }_1\right)\left(\gamma +{\mu }_1\right)\right],b=-{\Lambda }_2{\beta }_1{\beta }_2{\mu }_1^2{\lambda }_1{\lambda }_2\left({\lambda }_1+{\mu }_1\right)\left({\mu }_1+\omega \right)-2{\Lambda }_1{\mu }_1{\mu }_2^2\omega {\left({\lambda }_1+{\mu }_1\right)}^2\left({\lambda }_2+{\mu }_2\right)\left(\gamma +{\mu }_1\right)-{\Lambda }_1{\beta }_2{\mu }_1^2{\mu }_2{\lambda }_1\left({\lambda }_1+{\mu }_1\right)\left({\lambda }_2+{\mu }_2\right)\left({\mu }_1+\omega \right),\text{and}c={\Lambda }_1{\Lambda }_2{\beta }_1{\beta }_2{\mu }_1^2{\lambda }_1{\lambda }_2\left({\mu }_2+\omega \right)-{\Lambda }_1^2{\mu }_1^2{\mu }_2^2\left({\lambda }_1+{\mu }_1\right)\left({\lambda }_2+{\mu }_2\right)\left(\gamma +{\mu }_1\right).}$ The biologically meaningful root of this equation is given by (26)${\displaystyle E^{\ast }=\frac{-b-\sqrt{b^2-4ac}}{2a}.}$

Figure 6 shows the graphs of the number and proportion of symptomatically infectious individuals in the population as functions of the emigration rate ω. As more individuals leave the island, the overall population level declines, and hence there are fewer infected individuals. However, the infection spreads faster among the remaining inhabitants, resulting in a greater proportion of infected individuals in the population. The proportion of symptomatically infectious individuals grows with the migration rate, but it asymptotically approaches the value (27)${\displaystyle \underset{\omega \to \infty }{lim}\frac{I^{\ast }}{N^{\ast }}=\frac{\varphi {\mu }_1{\lambda }_1}{\left(\gamma +{\mu }_1\right)\left({\lambda }_1+{\mu }_1\right)}.}$

We next consider a game-theoretic model of individual migration decisions. Suppose that the population adopted the emigration rate ω_pop. A focal susceptible individual is presented with a choice to either migrate or not migrate. Each of the two strategic choices carries a corresponding payoff: E_m for migrate and E_nm for not migrate, given by (28)${\displaystyle E_{\text{m}}\left({\omega }_{\text{pop}}\right)=-C_b-{\omega }_{\text{pop}}{C}_s,\text{and}{E}_{\text{nm}}\left({\omega }_{\text{pop}}\right)=-\pi \left({\omega }_{\text{pop}}\right)C_i,}$ where C_b is the base (fixed) cost of migration, C_s is the scaling cost of migration, C_i is the cost of the (symptomatic) chikungunya infection, and π(ω_pop) is the probability of getting infected given the population emigration rate ω_pop. We assume that the cost of emigration is an increasing function of the migration rate because of the limited immigration potential of Mauritius: the more individuals migrate to Mauritius, the harder it becomes to find housing and jobs. For simplicity, we model the increasing emigration cost as a linear function of the migration rate. The probability of getting infected and incurring the cost of a symptomatic chikungunya infection if remaining on Reunion Island is the transition probability from the susceptible class S to the symptomatically infectious class I: (29)${\displaystyle \pi \left({\omega }_{\text{pop}}\right)=\frac{f_1\left({\omega }_{\text{pop}}\right)}{{\mu }_1+f_1\left({\omega }_{\text{pop}}\right)}\frac{\varphi {\lambda }_1}{{\mu }_1+{\lambda }_1}.}$ This probability is an increasing function of the emigration rate because each of the remaining susceptible individuals faces a higher risk of infection (cf. Fig. 5); the graph is shown in Fig. 7.

To find conditions when a focal susceptible individual should emigrate to Mauritius or remain on Reunion, we scale the payoffs in Eq. (28) by 1∕C_i and obtain (30)${\displaystyle E_{\text{m}}=-\widetilde{C_b}-{\omega }_{\text{pop}}\widetilde{C_s},\text{and}{E}_{\text{nm}}=-\pi \left({\omega }_{\text{pop}}\right),}$ where ${\tilde{C}}_b$ and ${\tilde{C}}_s$ are relative base and scaling costs of emigration, respectively. A susceptible individual should emigrate when the relative cost of doing so is less than the probability of getting infected: $\widetilde{C_b}+{\omega }_{\text{pop}}\widetilde{C_s}<\pi \left({\omega }_{\text{pop}}\right)$, and the individual should remain on the island otherwise. The regions in the $\left({\tilde{C}}_b,\omega \right)$-parameter space corresponding to the best choice for a focal individual for several values of ${\tilde{C}}_s$ are shown in Fig. 8. If the scaling cost of emigration C_s is negligible (i.e., the cost of emigration does not depend on the number of emigrating individuals), then the best strategy of a susceptible individual is to emigrate as long as the relative base cost of emigration $\widetilde{C_b}$ is sufficiently small (Fig. 8A). On the other hand, as the relative scaling cost of emigration $\widetilde{C_s}$ grows, the individual’s decision to emigrate starts to depend on the emigration decisions of other individuals (Figs. 8B–8C), until it becomes unprofitable to emigrate regardless of the relative base cost of emigration if the emigration rate is too high (Fig. 8D).

Optimal levels of mandatory emigration

Finally, we consider the potential impacts on the chikungunya epidemic on Reunion Island of coordinated emigration efforts. A mandating organization attempts to minimize overall costs, which are comprised of the cost of treatment of symptomatically infected individuals and the relocation costs of emigrating individuals. To estimate the number of emigrating susceptible individuals, we consider the difference between the total population size at equilibrium without emigration (N^∗ = Λ₁∕μ₁) and the total population size at equilibrium given the population migration rate ω (this expression is given in the first equation of (22)); we denote this difference by $N_{\omega }^{\ast }$.

The payoff of the emigration policy with migration rate ω is given by (31)${\displaystyle E\left(\omega \right)=-I^{\ast }-{\tilde{C}}_m{N}_{\omega }^{\ast },}$ where ${\tilde{C}}_m=C_m∕C_i$ is the cost of migration relative to the cost of infection. The graphs of this function for several values of ${\tilde{C}}_m$ are shown in Fig. 9. There are three qualitatively different outcomes:

1. For very low relative migration cost (${\tilde{C}}_m\le 0.00022$), higher migration rates result in smallest overall costs; however, the near-optimal costs are quickly achieved by small values of migration rate (ω = 0.01)—see Fig. 9A.

2. There is a small interval of the relative migration cost values ($0.00023\le {\tilde{C}}_m\le 0.00025$) where the optimal cost is achieved in the interior for very small values of the migration rate (ω < 0.002)—see Figs. 9B and 9C.

3. For all sufficiently large values of the relative migration cost (${\tilde{C}}_m\ge 0.00026$), it is best not to allow individuals to emigrate from the island—see Fig. 9D.

In practice, however, the cost of emigration (such as relocation from Reunion to Mauritius) is usually comparable to or higher than the cost of the symptomatic chikungunya infection. Therefore, the scenario shown in Fig. 9D is the most realistic one: it is best not to allow susceptible individuals to leave the island during the outbreak.