O3. Malaria SIR model with interventions

Modeling infectious disease dynamics is a crucial tool for public health scientists working to understand and control outbreaks of diseases like malaria. Mathematical models, particularly variations of the Susceptible-Infected-Recovered (SIR) framework, allow researchers to simulate how a disease spreads through a population over time. By incorporating key biological and epidemiological parameters – such as the transmission rate between humans and mosquitoes – these models provide a powerful means to predict the trajectory of an outbreak, estimate the total burden of disease, and, most importantly, evaluate the potential impact of different public health interventions.

SIR model for malaria

We will build our model off the vector-borne SIR model.  We will additionally account for births and deaths of both humans and mosquitoes and waning immunity in humans, since we will be modeling disease progression over months and years.

Figure 1:  SIR stocks and flows diagram for malaria accounting for births, deaths, and waning immunity. [Image Description]

For humans, we include the susceptible [latex](S_h)[/latex], infected [latex](I_h)[/latex], and recovered [latex](R_h)[/latex] sub-populations.  For mosquitoes, we assume that infected mosquitoes cannot recover, so we only include susceptible [latex](S_v)[/latex] and infected [latex](I_v)[/latex] sub-populations.  For both mosquitoes and humans, we consider all sub-populations of an organism equally likely to give birth with rate constants [latex]b_h[/latex] and [latex]b_v[/latex], and newborn organisms are susceptible.  Again, mosquitoes and humans from their respective sub-populations are equally likely to die due to causes other than malaria with rate constants [latex]d_h[/latex] and [latex]d_v[/latex].  We additionally include a sub-population [latex](D_h)[/latex] to track humans who have died due to disease but not due to natural causes with a mortality rate of [latex]m[/latex].  Finally, when modeling malaria over many years, we must account for waning immunity in humans, which occurs with a rate of [latex]w[/latex].  Now we will consider how to modify this model in order to account for interventions to curb the spread of malaria.

Decrease transmission

Interventions that decrease contact between humans and mosquitoes are critical for breaking the malaria transmission cycle. The most common examples include Insecticide-Treated Nets (ITNs), which physically and chemically protect sleeping humans from bites, and Spatial Repellents.

Spatial Repellents are devices or products that release volatile chemicals into the air, creating a zone that deters mosquitoes from entering an occupied space, such as a home. Unlike insecticides, which kill the vector, these products act primarily by repelling the mosquitoes, thus reducing the number of biting events.

In the SIR model, the effectiveness of these contact-reducing interventions is captured by introducing a reduction factor, [latex]c[/latex] (where [latex]0 \lt c \lt 1[/latex]), that scales down the original transmission rates. The transmission rate from mosquitoes to humans [latex](β)[/latex] becomes [latex]β_{eff}​=β(1−c)[/latex], and the rate from humans to mosquitoes [latex](α)[/latex] becomes [latex]α_{eff}​=α(1−c)[/latex]. By lowering both [latex]α[/latex] and [latex]β[/latex], these interventions drastically reduce the rate of new human infections and the number of mosquitoes that become infectious. Mathematically, this reduction helps to push the Basic Reproduction Number [latex](R_0​)[/latex] toward and below 1, which is the necessary condition for eliminating the disease from the population.

Decrease the mosquito population

Another powerful strategy for malaria control focuses on reducing the mosquito population [latex](N_v​=S_v+I_v)[/latex] itself, which limits the number of vectors available to transmit the disease. These interventions are modeled by manipulating the mosquito’s natural demographic rates: the death rate [latex](d_v​)[/latex] and the birth rate [latex](b_v​)[/latex]. Interventions that kill mosquitoes, such as using larvicides or adulticides, are modeled by increasing the natural death rate [latex](d_v​)[/latex]. We introduce an intervention-driven mortality rate, [latex]k[/latex], making the new effective death rate [latex]d_{v,eff}​=d_v​+k[/latex]. Conversely, techniques like the Sterile Insect Technique (SIT), which releases sterile males to reduce viable offspring, are modeled by decreasing the birth rate [latex](b_v​)[/latex]. We introduce a factor, [latex]s[/latex], for the fractional reduction in viable births, making the new effective birth rate [latex]b_{v,eff}​=b_v​(1−s)[/latex].

Both methods achieve the goal of drastically shrinking the total mosquito population [latex](N_v​)[/latex]. This reduction in [latex]N_v[/latex]​ directly leads to a smaller infected mosquito population [latex](I_v​)[/latex]. Since human infection incidence is proportional to [latex]I_v[/latex]​, a smaller vector population results in fewer infectious bites and thus fewer new human cases [latex](S_h​→I_h​)[/latex]. Mathematically, reducing [latex]N_v[/latex]​ is highly effective in lowering the Basic Reproduction Number [latex](R_0​)[/latex], providing a strong pathway toward malaria control and elimination.

Human vaccines

Vaccination is a powerful strategy to control malaria by modifying the human host’s susceptibility and disease course. Depending on the vaccine’s mechanism of action, it can be modeled in several ways.

The vaccine provides complete protection

In the case of sterilizing immunity, vaccinated individuals are prevented from ever becoming infected. This is modeled by taking a fraction of the Susceptible population [latex](S_h​)[/latex] and moving them directly into the Recovered population [latex](R_h​)[/latex], effectively giving them immediate, durable immunity. If we denote the rate of vaccination as [latex]v[/latex], the model adds a new flow: [latex]vS_h[/latex]​ from [latex]S_h[/latex]​ to [latex]R_h​[/latex].  This intervention could be modeled as a one-time or ongoing vaccine campaign, based on the considerations discussed in Chapter N4.

The vaccine reduces the likelihood of infection

Many current or candidate vaccines don’t completely prevent infection but make a susceptible person less likely to contract the disease upon being bitten by an infected mosquito. This is mathematically equivalent to reducing the transmission efficiency. This is modeled by decreasing the mosquito-to-human transmission rate [latex](β)[/latex]. If the vaccine provides a fractional protection [latex]p[/latex], the new effective transmission rate becomes [latex]β_{eff}​=β(1−p)[/latex].

The vaccine reduces the duration of illness

Some vaccines allow infection but lessen the severity, often by clearing the parasite faster. This is modeled by effectively increasing the recovery rate [latex](ν)[/latex] for vaccinated individuals, as they spend less time in the Infected stock [latex](I_h​)[/latex]. If the vaccination results in a faster clearance rate [latex]c[/latex], the new effective recovery rate is [latex]ν_{eff}​=ν+c[/latex].

All vaccination strategies aim to reduce the Infected human population [latex](I_h​)[/latex]. Moving people directly to [latex]R_h[/latex]​ or reducing [latex]β[/latex] lowers the incidence of new cases. Increasing [latex]ν[/latex] shortens the time individuals remain infectious, lowering the prevalence of the disease. In all three scenarios, a decrease in [latex]I_h[/latex]​ is the indirect mechanism for reducing onward transmission to susceptible mosquitoes [latex](αS_v​I_h​)[/latex]. Ultimately, a successful vaccination campaign substantially lowers [latex]I_h[/latex]​ and [latex]I_v[/latex]​, decreasing the Basic Reproduction Number [latex](R_0​)[/latex] and reducing malaria’s prevalence in the population.

Antimalarial drugs

Antimalarial drugs are a cornerstone of disease control, and their effects can be modeled by targeting multiple parameters within the SIR framework, depending on the drug’s mechanism of action. Broadly, these drugs can affect three stages of the parasite’s life cycle relevant to our model: preventing initial infection, stopping transmission, and treating the disease.

Preventing Initial Infection

Some drugs are used to prevent infection after a person is bitten, typically by killing the parasite’s early forms (sporozoites) before they can establish in the liver. For individuals using these drugs, the probability of successful transmission is reduced. This is modeled by decreasing the mosquito-to-human transmission rate [latex](β)[/latex]. If the drug provides a fractional protection [latex]p[/latex], the effective rate becomes [latex]β_{eff}​=β(1−p)[/latex].

Decreasing Transmission to Mosquitoes

Drugs that specifically target the sexual stages of the parasite (gametocytes) circulating in the human blood make an infected person less likely to pass the infection on to a biting mosquito. This reduces the human’s infectiousness. This effect is modeled by decreasing the human-to-mosquito transmission rate [latex](α)[/latex].  If the drug provides a fractional protection [latex]p[/latex], the effective rate becomes, similarly, [latex]α_{eff}​=α(1−p)[/latex].

Treating and Curing the Infection

Standard clinical treatment targets the blood-stage parasites that cause illness. This has two critical effects: decreasing disease mortality [latex](μ)[/latex] and decreasing the length of infection (i.e., decreasing [latex]L[/latex], which corresponds to an increase in the recovery rate [latex](ν)[/latex]). Decreasing [latex](μ)[/latex] ensures fewer individuals move to the [latex]D_h[/latex]​ stock, representing human deaths due to disease. Increasing [latex]ν[/latex] means people move faster from the [latex]I_h[/latex]​ stock to the [latex]R_h[/latex]​ stock, modeled as [latex]ν_{eff}​=ν+t[/latex], where [latex]t[/latex] is the treatment-induced recovery rate.

All these mechanisms – decreasing [latex]β[/latex], [latex]α[/latex], and [latex]μ[/latex], while increasing [latex]ν[/latex] – work synergistically to control the outbreak. By lowering the number of infected humans [latex](I_h​)[/latex] and reducing their infectiousness [latex](α)[/latex], antimalarial drugs drastically cut the transmission rate to the mosquito population [latex](I_v​)[/latex], which is essential for pushing the Basic Reproduction Number [latex](R_0​)[/latex] toward elimination.

Image Description

Figure 1:

The image is a flowchart diagram with two main sections representing disease dynamics in susceptible and infected populations. The upper section features two variables: (S_v) and (I_v), with arrows indicating transitions between states. A horizontal arrow labeled ( \alpha I_h S_v ) moves from (S_v) (left) to (I_v) (right). Both (S_v) and (I_v) have downward arrows labeled (b_v(S_v+I_v)) and (d_v I_v) respectively, indicating birth and death rates.

The lower section displays three variables: (S_h), (I_h), and (R_h). An arrow labeled ( \beta I_v S_h ) transitions from (S_h) to (I_h), with a branching arrow from (I_h) pointing to (D_h), labeled (m I_h). Another horizontal arrow labeled ( \nu I_h ) points from (I_h) to (R_h). Curved arrows indicate other transitions like (w R_h) returning back to (S_h). Downward arrows in each variable represent specific rates: (b_h(S_h+I_h+R_h)), (d_h S_h), (d_h I_h), and (d_h R_h).  [Return to Figure]

Licenses and Attributions

“Malaria SIR model with interventions” by Michelle McCully was written with assistance from Google Gemini 2.5 Flash.  “Malaria SIR model with interventions” is licensed under CC BY-NC 4.0.