High endemic levels of typhoid fever in rural areas of Ghana may stem from optimal voluntary vaccination behaviour

Abstract

Typhoid fever has long established itself endemically in rural Ghana despite the availability of cheap and effective vaccines. We used a game-theoretic model to investigate whether the low vaccination coverage in Ghana could be attributed to rational human behaviour. We adopted a version of an epidemiological model of typhoid fever dynamics, which accounted not only for chronic life-long carriers but also for a short-cycle transmission in the immediate environment and a long-cycle transmission via contamination of the water supply. We calibrated the model parameters based on the known incidence data. We found that unless the (perceived) cost of vaccination is negligible, the individually optimal population vaccination rate falls significantly short of the societally optimal population vaccination rate needed to reach herd immunity. We expressed both the herd immunity and the optimal equilibrium vaccination rates in terms of only a few observable parameters such as the incidence rate, demographics, vaccine waning rate and the perceived cost of vaccination relative to the cost of infection. This allowed us not to rely on other uncertain epidemiological model parameters and, in particular, to bypass uncertainties about the role of the carriers in the transmission.

1. Introduction

Typhoid fever is a systemic infection caused by the Salmonella enterica serotype Typhi (S. Typhi) bacteria [1]. The disease remains a serious public-health problem in developing countries; globally it causes between 11 and 21 million cases and between 128 000 and 161 000 deaths annually [2]. People are the only natural host and reservoir for S. Typhi [1]. The key preventive strategies are safe water, safe food, personal hygiene, and appropriate sanitation [2]. An additional effective tool for prevention of typhoid fever is vaccination [2]. The vaccines, however, are used sparingly, mostly because of the limited duration of the protection [3]. The vaccine is effective for about 3 years [4]. Revaccination is recommended every 3–7 years [2]. Also, lack of financing of typhoid fever vaccines has been a major bottleneck in nationwide adoptions [3].

There is a large number of mathematical models of typhoid fever dynamics, starting with Cvjetanović et al. [5] and including Cvjetanovic et al. [6], González-Guzmán [7], Lange et al. [8]; see for example Watson & Edmunds [9] and Bakach et al. [10] for comprehensive reviews. To accurately capture typhoid fever dynamics, the models include some version of the following compartments: susceptible, infected, chronic, recovered and vaccinated individuals; see for example Bailey [11]. Several models consider co-infections with other diseases such as malaria [12,13] or pneumonia [14]. Recent models such as Pitzer et al. [15,16] serve as policy recommendations for typhoid fever prevention and provide insights on the possible impact of vaccination. Mushayabasa [17] models the impact of screening and quarantine on typhoid fever control and elimination.

Starting with Bauch & Earn [18], game theory has been increasingly applied to help model prevention of diseases such as African trypanosomiases [19], chikungunya [20], cholera [21], dengue [22], Ebola [23], hepatitis B [24], hepatitis C [25], malaria [26], meningitis [27], monkeypox [28], polio [29] and toxoplasmosis [30]. In a recent review of game-theoretical models, Chang et al. [31] found that perceived risk and perceived vaccination costs, as opposed to social benefits, are key factors in determining the level of vaccination within a population.

In this paper, we use game theory to model typhoid fever immunization in Ghana. Ghana has a functioning and expanding healthcare system in place [32]. However, Ghana is scheduled to phase out of Gavi, the Vaccine Alliance, support by 2022 [33]. More than five million people (≈18%) do not have access to safe water and rely on surface water; over 80% of the population do not have access to improved sanitation/toilet facilities [34]. Typhoid fever treatments are available [35]; yet, the frequency of hospital attendance decreases constantly with increasing distance to the health facility [36]. Over a period of 8 years, typhoid fever surveillance has been conducted in the Asante Akim North District [37]. Incidence rates ranged from 120/100 000 person-years of observation (PYO) in 2007–2008 [38] to 389/100 000 PYO between 2010 and 2014 [39]; the incidence in the rural areas, 636/100 000 PYO, is more than twice the incidence in the urban areas. These data demonstrate how typhoid fever incidence rates vary in Ghana and hence pose a challenge to devise an effective typhoid fever control strategy for the country [35].

Our goal is to determine what vaccination rate is optimal from the individual perspective, and how the perceived cost of vaccination influences individual behaviour. We construct a game-theoretic model where susceptible individuals choose to vaccinate or not vaccinate against typhoid fever, and we find an equilibrium population vaccination rate when all individuals act rationally in their own self-interest. We are interested in comparing this individually selfish optimal vaccination rate to the herd immunity vaccination rate, which corresponds to the societal optimum.

In §2, we extend a version of the compartmental ODE model for typhoid fever dynamics with vaccination from Adetunde [40] and Mushayabasa [41]. Our model involves the ‘short-cycle’ transmission via contamination of food, drinking water, etc. in the immediate environment as done in Mushayabasa [41] as well as the ‘long-cycle’ transmission via contamination of the water supply as done in Pitzer et al. [15]. We estimate the epidemiological model parameters specifically for rural areas of Ghana in §2 based on the data from the current literature. We provide the ODE equilibria in §3, and use these solutions to estimate the overall effective transmission rate from the available typhoid fever incidence data. This allows us to bypass the uncertainty of the exact role of asymptomatic carriers in the disease transmission. We construct and solve a game-theoretic model of optimal individual vaccination decisions in §4. We conclude the paper by a discussion in §5.

2. Mathematical model of typhoid fever dynamics

We adopt a version of the model of typhoid fever dynamics with vaccination shown in figure 1, developed by Adetunde [40] and Mushayabasa [41]. The population is divided into five homogeneous compartments: susceptible (S), vaccinated (V), acutely infectious (I), carriers (C) and recovered (R) individuals. We also consider the ‘long-cycle’ transmission via contamination of the water supply (W) as done in Pitzer et al. [15].

Figure 1.

The scheme of the typhoid fever population dynamics, extending the model of the ‘short-cycle’ transmission via contamination of food, drinking water, etc. in the immediate environment as done in Mushayabasa [41] by a ‘long-cycle’ transmission via contamination of the water supply as done in Pitzer et al. [15].

All individuals die at a natural per capita annual mortality rate μ = 1/64 [42]. Individuals are born as susceptible at the same birth rate μ = 0.015625 per year. A susceptible individual can acquire an infection via contact with an infected individual, carrier, or through the infected water supply. The force of infection is given by α_I I + α_C C + α_W W, where α_I, α_C, α_W are the effective transmission rates. We will see in §3 that the system reduces to the system from Mushayabasa [41] with α_I I + α_C C + α_W W replaced by αI. We estimate the value of α in §3c from the known annual incidence rate i = 0.00636 (636/100 000 PYO from Marks et al. [39]) in rural areas of Ghana.

As in Adetunde [40], Mushayabasa [41] and Pitzer et al. [15], we ignore the incubation period of 10–14 days and assume that susceptible individuals directly become infectious. With the introduction of early and appropriate antibiotic therapy, the case fatality rates for typhoid fever are typically less than 1% [43]. Assuming the average fatality to be 1%, the disease-induced mortality rate, which we will denote σ, solves σ/(σ + μ) = 0.01, and hence σ = 1.58 × 10⁻⁴ per year. We will therefore neglect σ in our model completely. If treated, the infectious individuals recover in 3–7 days [43]. Without treatment, the fever lasts 4–6 weeks [1,2]. We will assume that the average duration of the acute stage is 7 days.

Not all individuals fully recover. Instead, about 2–5% of the patients become chronic (potentially lifelong) carriers [44,45]. We will assume that 96.5% of the patients with acute infection fully recover and 3.5% become carriers. This means that the recovery rate from the acute infection is b = 0.965 × 365/7 ≈ 50 per year, and the rate at which infectious individuals become carriers is β = 0.035 × 365/7 = 1.825 per year. The carriers are asymptomatic and we assume no disease-induced mortality in the carriers because the mortality is lower than the acute-fever-induced mortality. The carriers recover at the rate γ = 0.04 per year [11]. Reinfections with typhoid fever are rare [46], and hence we assume that recovered individuals gain a lifelong immunity to the disease.

Susceptible individuals vaccinate at the rate ϕ. We assume that the vaccine offers complete protection (i.e. a vaccinated individual does not get infected). On the other hand, we will assume that the vaccine protection wanes at the rate ω = 1/3 per year [4].

As in Pitzer et al. [15], we assume that infectious individuals and carriers contaminate water supply by shedding bacteria at the rates s_I and s_C, respectively. The infectious particles in the water decay at rate ξ.

3. Analysis of the epidemiological model

The typhoid fever transmission dynamics presented in figure 1 can be described by the following system of differential equations:

$$\begin{array}{rl}& \frac{\text{d}S}{\text{d}t}=\mu +\omega V-(\mu +\varphi +{\alpha }_{I}I+{\alpha }_{C}C+{\alpha }_{W}W)S,\end{array}$$ 3.1

$$\begin{array}{rl}& \frac{\text{d}V}{\text{d}t}=\varphi S-(\mu +\omega )V,\end{array}$$ 3.2

$$\begin{array}{rl}& \frac{\text{d}I}{\text{d}t}=({\alpha }_{I}I+{\alpha }_{C}C+{\alpha }_{W}W)S-(\mu +\beta +b)I,\end{array}$$ 3.3

$$\begin{array}{rl}& \frac{\text{d}C}{\text{d}t}=\beta I-(\mu +\gamma )C,\end{array}$$ 3.4

$$\begin{array}{rl}& \frac{\text{d}R}{\text{d}t}=\gamma C+bI-\mu R\end{array}$$ 3.5

$$\begin{array}{rl}\text{and}& \frac{\text{d}W}{\text{d}t}=s_{I}I+s_{C}C-\xi W.\end{array}$$ 3.6

(a). Relationship to model of Mushayabasa [41]

In an equilibrium, one has

$$C=\frac{\beta }{\mu +\gamma }I$$ 3.7

and

$$W=\frac{1}{\xi }(s_{I}I+s_{C}C)=\frac{1}{\xi }(s_I+s_C\frac{\beta }{\mu +\gamma })I.$$ 3.8

Thus, by the theory of asymptotically autonomous systems [47,48], one can replace equations (3.1) and (3.3) by

$$\frac{\text{d}S}{\text{d}t}=\mu +\omega V-(\mu +\varphi +\alpha I)S$$ 3.9

and

$$\frac{\text{d}I}{\text{d}t}=\alpha SI-(\mu +\beta +b)I,$$ 3.10

where

$$\alpha ={\alpha }_I+{\alpha }_C\frac{\beta }{\mu +\gamma }+{\alpha }_W\frac{1}{\xi }(s_I+s_C\frac{\beta }{\mu +\gamma }).$$ 3.11

The resulting system (3.9), (3.10), (3.2), (3.4) and (3.5) is a special case of the model of Mushayabasa [41]; in their notation, here we assume that Λ = μ, σ = 0, and d = 0.

(b). Equilibria of the dynamics

There are two equilibria of the dynamics (3.1)–(3.6): the disease-free equilibrium E₀ = (S₀, V₀, I₀, C₀, R₀, W₀) and the endemic equilibrium E* = (S*, V*, I*, C*, R*, W*).

The disease-free equilibrium E₀ = (S₀, V₀, I₀, C₀, R₀, W₀) satisfies 0 = I₀ = C₀ = R₀ and thus W₀ = 0 and

$$S_0=\frac{\omega +\mu }{\omega +\mu +\varphi }$$ 3.12

and

$$V_0=\frac{\varphi }{\omega +\mu +\varphi }.$$ 3.13

The effective reproduction number R₀(ϕ) depends on the population vaccination rate ϕ, and it was computed in Mushayabasa [41] using the next generation matrix approach [49]. Hence, for our model,

$$R_0(\varphi )=\frac{\omega +\mu }{\omega +\mu +\varphi }\cdot \frac{\alpha }{\mu +\beta +b}.$$ 3.14

One can also obtain this expression by using the definition of the basic reproduction number as the number of secondary infections caused by a single infected individual in a disease-free population. An individual stays infected for a period 1/(μ + β + b). During that period they infect susceptible individuals at the rate α_I S₀ and shed bacteria at the rate s_I. With probability β/(μ + β + b), the infected individual becomes a carrier. The individual stays in the carrier state for a period 1/(μ + γ) during which they infect susceptible individuals at the rate α_C S₀ and shed bacteria at the rate s_C. The infectious particles stay in the water for a period 1/ξ during which they infect susceptible individuals at the rate α_W S₀. Putting it all together yields (3.14). The disease-free equilibrium is stable when R₀(ϕ) < 1 [41].

The endemic equilibrium E* = (S*, V*, I*, C*, R*, W*) is given by

$$\begin{array}{rl}{S}^{\ast }& =\frac{\beta +b+\mu }{\alpha }=\frac{S_0}{R_0(\varphi )},\end{array}$$ 3.15

$$\begin{array}{rl}{V}^{\ast }& =\frac{\varphi }{\omega +\mu }\cdot \frac{\beta +b+\mu }{\alpha }=\frac{V_0}{R_0(\varphi )},\end{array}$$ 3.16

$$\begin{array}{rl}{I}^{\ast }& =\frac{\mu }{\alpha }\cdot \frac{\omega +\mu +\varphi }{\omega +\mu }(R_0(\varphi )-1)=\frac{\mu }{\alpha S_0}(R_0(\varphi )-1),\end{array}$$ 3.17

$$\begin{array}{rl}{C}^{\ast }& =\frac{\beta }{\gamma +\mu }{I}^{\ast },\end{array}$$ 3.18

$$\begin{array}{rl}{R}^{\ast }& =\frac{1}{\mu }\cdot (b+\frac{\beta \gamma }{\gamma +\mu })I^{\ast }\end{array}$$ 3.19

$$\begin{array}{rl}\text{and}{W}^{\ast }& =\frac{1}{\xi }(s_I+s_C\frac{\beta }{\mu +\gamma })I^{\ast }.\end{array}$$ 3.20

The endemic equilibrium is stable when R₀(ϕ) > 1 [41]. The detailed calculations of the equilibria values are shown in appendix A.

(c). Estimating the effective transmission rate

We will estimate the value of the parameter α for the overall effective transmission rate assuming the typhoid fever transmission dynamics has reached an endemic equilibrium. The per capita incidence rate (i.e. the per capita number of new cases annually) is given by i = αI* S* [47,50]. It follows from (3.15) and (3.17) that $i=\mu (1-\frac{1}{R_0(\varphi )})$ and thus, by (3.14),

$$\alpha =(\mu +\beta +b)\cdot \frac{(\omega +\mu +\varphi )}{\omega +\mu }\cdot \frac{\mu }{\mu -i}.$$ 3.21

As noted in Date et al. [51] and Slayton et al. [52], the current typhoid fever vaccination rate, ϕ₀, is very low in rural Ghana. For the purpose of estimating the value of α, we will therefore assume that ϕ₀ = 0. Using the parameter values from table 1, including the incidence rate i = 0.00636 [39], we obtain α ≈ 87.7. Figure 2 illustrates the estimates of α for several different values of ϕ₀.

Table 1.

Summary of the parameters of the typhoid fever transmission dynamics model. All rates are per capita per year. The outcomes of the game-theoretic model depend only on μ, i, ϕ₀, ω and K.

symbol	meaning	base value	reference(s)
μ	natural mortality rate	1/64	World Bank [42]
i	incidence rate	0.00636	Marks et al. [39]
ϕ0	current vaccination rate	≈0	Date et al. [51]
ω	vaccine waning rate	1/3	Klugman et al.[4]
K	perceived vaccination cost relative to the infection cost	varies	Antillón et al. [53]
α	overall effective transmission rate	87.7	§3c
αI	effective transmission rate from infected	unknown
αC	effective transmission rate from carriers	unknown
αW	effective transmission rate from water supply	unknown
sI	rate of shedding from infectious individuals	1	Pitzer et al. [15]
sC	rate of shedding from carriers	0.01	Pitzer et al. [15]
ξ	decay rate of infectious particles in water	17	Cho & Kim [54]
b	recovery rate for infectious individuals	50	Parry et al. [43]
β	rate of becoming a carrier	1.825	Gunn et al. [43], Parry et al. [44]
γ	carrier recovery rate	0.04	Bailey [11]
ϕ	vaccination rate	varies

Figure 2.

Estimating the overall effective transmission rate, α, from the known incidence rate, i. The transmission rate α depends on the vaccination rate ϕ₀ when the incidence rate was obtained, ϕ₀ = 0 (solid line), ϕ₀ = 0.03 (dotted line) and ϕ₀ = 0.06 (dashed line).

It follows from (3.21) and (3.14) that

$$R_0(\varphi )=\left(\frac{\mu +\omega +{\varphi }_0}{\mu +\omega +\varphi }\right)\left(\frac{\mu }{\mu -i}\right).$$ 3.22

In particular, we can estimate R₀(ϕ) by knowing only i, ϕ₀, ϕ, and μ while not having to know α_I, α_C, α_W, s_I, s_C, ξ, b, β and γ. We note that regardless what the value of ϕ₀ is, R₀(ϕ₀) = μ/(μ − i) ≈ 1.69, and we take this value as the baseline estimate of the basic reproduction number.

(d). Herd immunity vaccination rate

The optimal vaccination rate for the society is the least vaccination rate when the population achieves herd immunity; we will denote this vaccination rate by ϕ_HI. Herd immunity results in a disease-free equilibrium (i.e. R₀(ϕ) ≤ 1). It follows that ϕ_HI = min{ϕ ∈ (0, ∞);R₀(ϕ) ≤ 1}. If R₀(0) < 1, then the herd immunity is achieved even without vaccination (i.e. ϕ_HI = 0). If R₀(0) > 1, then ϕ_HI is the solution to the equation R₀(ϕ_HI) = 1. It thus follows from (3.14) that

$${\varphi }_{\text{HI}}=max\{R_0(0)-1,0\}\cdot (\omega +\mu ).$$ 3.23

Figure 3 illustrates the effective reproduction number as a function of the population vaccination rate, and the value of ϕ_HI. For the parameter values as in table 1, ϕ_HI = 0.2373 ≈ 1/4.21, that is, to eradicate typhoid fever, the susceptible individuals should be vaccinating approximately once every 4 years. We note that the World Health Organization (WHO) recommends revaccination every 3 years for ViPS vaccine, and every 3–7 years in most endemic settings for Ty21a vaccines [2].

Figure 3.

Effective reproduction number as a function of the population vaccination rate ϕ. With the parameter values as in table 1, the smallest vaccination rate to achieve herd immunity is ϕ_HI = 0.2373 ≈ 1/4.21, that is, the susceptible individuals should vaccinate about once every 4 years.

Moreover, by (3.22) (assuming i > 0),

$${\varphi }_{\text{HI}}=(\mu +\omega +{\varphi }_0)\frac{\mu }{\mu -i}-(\mu +\omega ).$$ 3.24

In particular, we can estimate the vaccination level needed for the immunity when we know only i, ϕ₀, μ, and ω but not the other epidemiological model parameters.

4. Game-theoretic model of voluntary vaccination decisions

In this section, we set up and solve a game-theoretic model of individual vaccination decisions against typhoid fever. The players in this vaccination game are susceptible individuals, and there are two strategies: to vaccinate or not to vaccinate. We assume that all individuals are rational and act in their own self-interest [55]. Individuals weigh the (perceived) cost of vaccination versus the risk of infection [56]. The risk of the infection depends on the population vaccination rate, and this results in strategic interactions within the population because susceptible individuals base their vaccination decisions on the strategic choices of other individuals [18].

Let K be the perceived cost of vaccination relative to the perceived cost of infection. The actual vaccine cost is $1 per dose [53], and we take this as a base value for the cost of vaccination. We note, however, that the monetary cost of the vaccine is just one kind of direct cost associated with vaccination. Even if the vaccine is provided for free, there are other costs such as time loss and travel cost that all are particularly relevant in rural and poor areas [3]. Moreover, the indirect cost of vaccination may be influenced by the public perception of the morbidity risks due to vaccination and of the efficacy of the vaccine. While adverse vaccine effects are rare [57], vaccine scares or distrust of government institutions often increase the perceived cost of vaccination in the population. We also consider the treatment cost as the main indicator of the direct cost of infection. This cost depends on the location: it can be as low as $2 (in Kolkata or Lwak) or as high as $755 (in New Delhi) [53]. Additionally, the indirect cost of infection is based on the perceived morbidity risk of the disease in the population. We will use the value K = 1/200 = 0.005 as a rough baseline estimate of this game-theoretic model parameter.

We assume that the dynamics of typhoid fever has reached an endemic equilibrium [39]. In the endemic equilibrium, when a susceptible individual decides not to vaccinate, they can contract typhoid fever with the probability αI*/(αI* + μ), which is the transition probability from the compartment S to the compartment I. Since we use the dimensionless parameter K to represent the cost of vaccination relative to the cost of infection, the expected individual cost of not vaccinating is given by the probability of getting infected in the population where everyone else vaccinates at the rate ϕ:

$$C_{\text{NV}}(\varphi )=\frac{\alpha I^{\ast }}{\alpha I^{\ast }+\mu }.$$ 4.1

This quantity depends on the population vaccination rate ϕ because the value of I* depends on ϕ via (3.17). If a susceptible individual decides to vaccinate, they are not guaranteed full protection from typhoid fever because the vaccine wanes off at the rate ω. Consequently, the expected cost of vaccinating when everyone else vaccinates at the rate ϕ is

$$C_{\text{V}}(\varphi )=\frac{\omega }{\omega +\mu }{C}_{\text{NV}}(\varphi )+K.$$ 4.2

We note that this computation involves a short-term estimate of the probability of getting infected of an individual that vaccinated once. In particular, we do not consider lifelong vaccination strategies (with possible revaccinations) because both the perceived cost of vaccination and the prevalence of the disease may change over time, and thus it is difficult to accurately estimate the expected costs of such long-term strategies.

If C_V > C_NV, then a focal individual does better by not vaccinating, and hence the population vaccination rate would tend to go down. If C_V < C_NV, then a focal individual does better by vaccinating, and hence the population vaccination rate would tend to go up. It follows that the equilibrium population vaccination rate ϕ_NE is the solution to the equation

$$C_{\text{V}}({\varphi }_{\text{NE}})=C_{\text{NV}}({\varphi }_{\text{NE}})$$ 4.3

and

$$\frac{\omega }{\omega +\mu }\frac{\alpha I^{\ast }}{\alpha I^{\ast }+\mu }+K=\frac{\alpha I^{\ast }}{\alpha I^{\ast }+\mu },$$ 4.4

which yields

$$\omega \alpha I^{\ast }+\alpha I^{\ast }K(\omega +\mu )+\mu K(\omega +\mu )=\alpha I^{\ast },$$ 4.5

and consequently,

$$I^{\ast }=\frac{K\mu (\mu +\omega )}{\alpha (\mu -K(\mu +\omega ))}.$$ 4.6

By (A 22),

$$\begin{array}{rl}{\varphi }_{\text{NE}}& =(\omega +\mu )(R_0(0)-1-\frac{(\omega +\mu )K}{\mu -(\omega +\mu )K})\\ & =(\omega +\mu )(R_0(0)-\frac{\mu }{\mu -(\omega +\mu )K}).\end{array}$$ 4.7

Furthermore, using (3.22), and making sure the rates are positive, we get

$${\varphi }_{\text{NE}}=\{\begin{array}{ll}(\omega +\mu )[(\frac{\mu +\omega +{\varphi }_0}{\mu +\omega })(\frac{\mu }{\mu -i})-\frac{\mu }{\mu -(\omega +\mu )K}],& K<K_{\text{max}}\\ 0,& K\ge K_{\text{max}},\end{array}$$ 4.8

where

$$\begin{array}{rl}{K}_{\text{max}}& =\frac{\mu }{\mu +\omega }(1-\frac{1}{R_0(0)})\\ & =\frac{\mu }{\mu +\omega }(1-\left(\frac{\mu +\omega }{\mu +\omega +{\varphi }_0}\right)\left(\frac{\mu -i}{\mu }\right))\end{array}$$ 4.9

is the threshold relative vaccination cost after which no rational individuals would vaccinate.

Figure 4 shows the equilibrium vaccination rate ϕ_NE as a function of the relative vaccination cost K. This is the individually optimal vaccination rate, and it falls below the societally optimal vaccination rate ϕ_HI unless K = 0, that is, unless the perceived cost of vaccination is negligible (zero mathematically).

Figure 4.

The optimal equilibrium vaccination rate ϕ_NE as a function of the relative vaccination cost K. The optimal vaccination rate ϕ_NE (and the herd immunity ϕ_HI) depends on the vaccination rate ϕ₀ when the incidence rate was obtained, ϕ₀ = 0 (solid line), ϕ₀ = 0.03 (dotted line) and ϕ₀ = 0.06 (dashed line). Other parameter values are as in table 1. The optimal vaccination rate ϕ_NE falls significantly short of the herd immunity vaccination rate ϕ_HI unless the perceived cost of vaccination is negligible. If ϕ₀ = 0, individuals will not vaccinate at all when the relative cost of vaccination exceeds the threshold value K_max ≈ 0.0184. If ϕ₀ = 0.6, K_max ≈ 0.0224.

5. Conclusions and discussion

In this paper, we investigated whether the low vaccination coverage against typhoid fever in rural areas of Ghana could be attributed to rational behaviour of selfish individuals. We adapted a model of typhoid fever transmission dynamics with vaccination from Mushayabasa [41] and included chronic lifelong carriers, a short-cycle transmission in the immediate environment, and a long-cycle transmission via contamination of the water supply as in Pitzer et al. [15,16]. We calibrated the model parameters based on available incidence data. This epidemiological model allowed us to find the population vaccination rate necessary to reach herd immunity ϕ_HI (see figure 3). Our model predicted that to reach herd immunity, susceptible individuals should vaccinate slightly less frequently than once every 4 years, which is consistent with the WHO recommendation of revaccination every 3 years for ViPS, and every 3–7 years in most endemic settings for Ty21a vaccines [2].

We then constructed a vaccination game, where the players are susceptible individuals. The individuals had two strategies in the game: to vaccinate or not to vaccinate. An individual who decided to vaccinate incurred the cost of the vaccine, yet they could still become infected if the vaccine waned. An individual who decided not to vaccinate incurred the cost of typhoid fever infection if the individual got infected, and the probability of getting infected depended on the vaccination decisions of the rest of the population. The resulting strategic interaction within the population led towards a population vaccination rate ϕ_NE, which is the equilibrium solution of the vaccination game. This equilibrium solution of the vaccination game can be thought of as optimal vaccinating behaviour of rational individuals who seek to maximize their own well-being only.

We expressed the equilibrium vaccination rate ϕ_NE as a function of the cost of vaccination relative to the cost of infection (relative vaccination cost) K, and compared it with the herd immunity vaccination rate ϕ_HI (see figure 4). We found that the optimal vaccination rate ϕ_NE falls significantly short of the herd immunity vaccination rate ϕ_HI unless the cost of vaccination is negligible. The vaccination cost is relatively high in rural areas of Ghana because Krumkamp et al. [36] show that hospitalization decreases with distance from the hospital, suggesting that travel costs play a substantial role in the individual perceived cost of vaccination. Therefore, the prediction of our game-theoretic model matches with the current endemic situation of typhoid fever in rural Ghana.

Our result for typhoid fever qualitatively agrees with previous findings for polio vaccination [18,29] and Hepatitis b vaccination [24], yet it disagrees with results for chikungunya [20], Ebola [23], malaria [26], dengue [22] or cholera [21] where the equilibrium population rate of adoption of a personal protective measure was very close to the herd immunity one.

The extensive review done by Watson & Edmunds [9] concluded that vaccines alone are unlikely to eliminate endemic disease in the short to medium term without measures to reduce transmission from asymptomatic carriage. It is also of interest to identify and evaluate other preventative measures such as behavioural modification in addition to vaccination that would have a measurable effect on the transmission of typhoid fever as it was in the case of cholera prevention [21].

One of the major strengths of our mathematical model was that the simplicity of the original model of Mushayabasa [41] allowed us to estimate the transmission rate (and the force of infection) directly from the disease incidence data without knowing too many details about the role of chronic carriers in the transmission dynamics. The precise role of carriers in disease transmission remains unclear; they presumably act as reservoirs for a diverse range of S. Typhi strains and may act as a breeding ground for new genotypes [58]. Our model can be further improved by incorporating the seasonality in the contamination of the water supply as done in Pitzer et al. [15,16].

Following Fu et al. [59], many recent studies such as Arefin et al. [60,61], Huang et al. [62], Iwamura & Tanimoto [63], Kabir & Tanimoto [64], and Kabir et al. [65,66] used multi-agent-simulation methodology, thus allowing considerable flexibility and realism in the modelling approach, especially in prevention of seasonal influenza where individuals have to revaccinate every year. Furthermore, complex networks provide a platform to model real-world populations [67]. The departure from a well-mixed population models brings heterogeneity in vaccinating actions [59]. The individuals with many contacts may prefer to voluntarily vaccinate in order to reduce the risk of being infected which can largely inhibit the outbreaks [68]. Given the relatively short duration of the protection of the typhoid fever vaccine, and consequently the potential need for regular revaccination, we believe that the methods developed in the aforementioned papers can be adapted for typhoid fever as well.

Appendix A. Equilibria of the dynamics

We have to solve the following system of (algebraic) equations:

$$\begin{array}{rl}0& =\mu +\omega V-(\mu +\varphi +{\alpha }_{I}I+{\alpha }_{C}C+{\alpha }_{W}W)S,\end{array}$$ A 1

$$\begin{array}{rl}0& =\varphi S-(\mu +\omega )V,\end{array}$$ A 2

$$\begin{array}{rl}0& =({\alpha }_{I}I+{\alpha }_{C}C+{\alpha }_{W}W)S-(\mu +\beta +b)I,\end{array}$$ A 3

$$\begin{array}{rl}0& =\beta I-(\mu +\gamma )C,\end{array}$$ A 4

$$\begin{array}{rl}0& =\gamma C+bI-\mu R\end{array}$$ A 5

$$\begin{array}{rl}\text{and}0& =s_{I}I+s_{C}C-\xi W.\end{array}$$ A 6

(a) Disease-free equilibrium

In the disease free equilibrium when 0 = I₀ = C₀ = R₀ = W₀, the system (A 1)–(A 6) reduces to

$$0=\mu +\omega V_0-(\mu +\varphi )S_0$$ A 7

and

$$0=\varphi S_0-(\mu +\omega )V_0.$$ A 8

Adding (A 7) and (A 8) together yields

$$S_0+V_0=1.$$ A 9

By (A 8),

$$V_0=\frac{\varphi }{\mu +\omega }{S}_0$$ A 10

and thus

$$S_0=\frac{\mu +\omega }{\mu +\omega +\varphi }$$ A 11

and

$$V_0=\frac{\varphi }{\mu +\omega +\varphi }.$$ A 12

(b) Endemic equilibrium

From (A 4), (A 3), (A 5) and (A 6), we get

$$\begin{array}{rl}{C}^{\ast }& =\frac{\beta }{\mu +\gamma }{I}^{\ast },\end{array}$$ A 13

$$\begin{array}{rl}{R}^{\ast }& =\frac{1}{\mu }(b+\frac{\beta \gamma }{\gamma +\mu })I^{\ast }\end{array}$$ A 14

$$\begin{array}{rl}\text{and}{W}^{\ast }& =\frac{1}{\xi }(s_I{I}^{\ast }+s_C{C}^{\ast })=\frac{1}{\xi }(s_I+s_C\frac{\beta }{\mu +\gamma })I^{\ast }.\end{array}$$ A 15

Thus, (A 1) and (A 3) reduce to

$$0=\mu +\omega V-(\mu +\varphi +\alpha I)S$$ A 16

and

$$0=\alpha SI-(\mu +\beta +b)I,$$ A 17

where

$$\alpha ={\alpha }_I+{\alpha }_C\frac{\beta }{\mu +\gamma }+{\alpha }_W\frac{1}{\xi }(s_I+s_C\frac{\beta }{\mu +\gamma }).$$ A 18

From (A 17), assuming I* ≠ 0,

$$S^{\ast }=\frac{\beta +b+\mu }{\alpha }.$$ A 19

From (A 2),

$$V^{\ast }=\frac{\varphi }{\omega +\mu }{S}^{\ast }$$ A 20

and finally, by (A 16)

$$\begin{array}{rl}{I}^{\ast }& =\frac{\mu +\omega V^{\ast }}{\alpha S^{\ast }}-\frac{\varphi +\mu }{\alpha }\\ & =\frac{\mu }{\beta +b+\mu }+\frac{\omega \varphi }{\alpha (\omega +\mu )}-\frac{\varphi +\mu }{\alpha }\\ & =\frac{\mu }{\beta +b+\mu }-\frac{\mu }{\alpha (\omega +\mu )}(\mu +\omega +\varphi )\\ & =\frac{\mu }{\alpha }\left(\frac{\mu +\varphi +\omega }{\mu +\omega }\right)(R_0(\varphi )-1).\end{array}$$ A 21

Note that it also follows from (A 21) that

$$\begin{array}{rl}\varphi & =\frac{\alpha (\omega +\mu )}{\mu }(\frac{\mu }{\beta +b+\mu }-I^{\ast })-(\omega +\mu )\\ & =(\omega +\mu )(R_0(0)-1-\frac{\alpha }{\mu }{I}^{\ast }).\end{array}$$ A 22

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Authors' contributions

H.O. and I.V.E. designed the study. C.A. and A.L. conducted the research under the supervision of H.O., I.V.E. and J.R. The final analysis was done by I.V.E., J.R. and D.T., and the manuscript was written by I.V.E., J.R. and D.T. The order of authorship was determined alphabetically.

Competing interests

There are no competing interests.

Funding

This research was conducted as part of a Research Experiences for Undergraduates programme at the University of North Carolina at Greensboro in summer 2017, which was funded by NSF grant no. DMS-1659646.