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# Modeling the Co-infection of Hepatitis E and Malaria in Uganda

Overcrowding in Uganda from increased immigration activity is causing insufficient sanitation and poor nutrition in many regions of the country. Heightened movement in, out, and around Uganda also facilitates the outbreak and spread of multiple diseases, making epidemiological mathematics more important than ever. One such disease is hepatitis E, an inflammation of the liver caused by a virus of the same name.

A hepatitis E outbreak is typically self-limiting, unless it exists in tandem with another disease. Co-infection occurs when a population harbors two infections simultaneously. For this reason, the 2007-2009 outbreak of hepatitis E in Kitgum, a municipality in northern Uganda was particularly noteworthy. Malaria is endemic throughout Uganda, and its presence complicates the behavior and handling of hepatitis E. While no real treatment exists for hepatitis E other than improved hygiene, it does impact the effectiveness of the malaria drug.

During a scientific session at the 9th International Congress on Industrial and Applied Mathematics, currently taking place in Valencia, Spain, Betty Nannyonga of Makerere University described a simple model to depict the role of malaria in Kitgum’s 2007-2009 hepatitis E outbreak. Her ultimate goal was to yield policy suggestions for the ministry’s management of the disease—which was attributed to a dearth of both latrines and safe drinking water—and estimate the number of latrines and boreholes necessary to stop the outbreak. Malaria weakens the immune system, especially in the concurrent presence of other infections. “We wanted to see how these two infections affected this region of the country, and what the ministry has to do to combat it,” Nannyonga said.

At ICIAM19, Betty Nannyonga described a simple model to depict the role of malaria in Kitgum’s 2007 hepatitis E outbreak.

Approximately 9,449 cases of hepatitis—out of a population of 28,045 in 6,039 households—were reported in Kitgum during the outbreak, resulting in 160 deaths. To address the co-infection of malaria and hepatitis E, Nannyonga developed two models: one depicting only hepatitis E with a constant population, and one depicting the disease’s co-infection with malaria. The latter model accounts for a changing population due to malaria’s endemic nature. She fit her models to the existing data using two approaches: linear regression (under the assumption that $$\mu =0$$) and a nonlinear differential equations fitting tool that investigates $$\mu>0$$. Based on the fitting, she estimated the transmission rate and basic reproduction number ($$R_0$$).

Without the presence of malaria, each person with hepatitis will likely affect two other individuals (with an $$R_0$$ level of 2.11). Nannyonga then turned to a mathematical modeling tool called PottersWheel to investigate how a fluctuating population changes the $$R_0$$. Because malaria is a continuous process, she used nonlinear ordinary differential equations to depict the varying ebb and flow of the population. “If you have malaria, what condition will facilitate hepatitis E, and the other way around?” she asked.

Using PottersWheel, Nannyonga set the fits in a sequence to 20, chose parameters, and ran the process 50s time for each value. The results of the fitting tool were substantial. “When you have malaria in your system, the chances of you ending up with hepatitis E is almost seven times greater than if you don’t have malaria,” Nannyonga said.

Next, she conducted cost-effective analysis of hepatitis E, beginning with an investigation of the disability-adjusted life year (DALY) — an indicator that quantifies a disease’s burden as well as the associated functional limitation and premature mortality. In short, this metric investigates the present and future cost of a disease for the affected country. “The DALY can be used across cultures to measure health gaps as opposed to health expectancies, and the difference between a current and ideal situation where everyone lives up to the age of the standard life expectancy in perfect health,” Nannyonga said. “It combines in one measure the time lived with the disability (YLD) and the time lost due to premature mortality (YLL).” Thus, $$DALY = YLD + YLL$$. More specifically, the equation breaks down in the following way:

• $$YLL=$$ number of deaths $$(P) \times$$ standard life expectancy $$(L)$$ at the age at which death occurs. $$YLL= P \times L$$.
• $$YLD =$$ number of disability cases $$(I) \times$$ the average duration of the disease $$(D) \times$$ a weight factor $$(DW)$$ that reflects the disease’s severity on a scale of $$0$$ (perfect health) to $$1$$ (dead). $$YLD = I \times DW \times D$$.

To estimate the net present value of years of life lost, Nannyonga applied a standard time discount rate to years of life lost in the future; this adjusts both the costs and health outcomes.

Finally, she turned to policymaking. There is currently one latrine per six households in Kitgum, and one borehole for every 263 households. To combat hepatitis E in the absence of malaria, one would have to increase latrine coverage to 16 percent from the current 3.7 percent, and borehole coverage to 17 percent from 0.38 percent. Malaria’s presence in in the population demands latrine coverage of 17.5 percent, with borehole coverage of 18.1 percent.

Next, Nannyonga calculated the cost of constructing additional latrines. There are currently 1,038 latrines in Kitgum, meaning that 3,477 additional latrines would be necessary to avoid future hepatitis E outbreaks. The estimated cost of one basic pit latrine is $250. Thus, building the required 3,477 latrines would cost$869,250. In this case, the averted cost per DALY is \$123.

While Nannyonga admits that her fitted models are fairly basic and may not fully reflect the complexity of either hepatitis E or malaria—in part due to a scarcity of credible data—fitting mathematical models to data is a great approach for exploring possible ways to eradicate diseases. “Our results highlight that protection against diseases is not only essential in terms of improving people’s lives, but also an important means of bolstering economic and social development,” she said.

 Lina Sorg is the associate editor of SIAM News.